MATCH MATCH Commun. Math. Comput. Chem.

79 (2018) 5-54
Communications in Mathematical
and in Computer Chemistry ISSN 0340 - 6253

New Geometric-Arithmetic Indices

Piotr Wilczek
Computer Laboratory, Na Skarpie 99/24, 61-163 Poznań, POLAND
piotr.wilczek.net@onet.pl
(Received May 22, 2017)
Abstract

Based on the definition of the general geometric-arithmetic index, this article introduces several new
geometric-arithmetic indices (Part One). In Part Three, we have determined the degree of degeneracy
of these new invariants and we have designated the first pairs (or subsets) of molecular graphs having
the same values of a given geometric-arithmetic index. It appears that in many cases the newly
proposed descriptors have a much greater level of uniqueness that the strongly discriminating Balaban
J index. In Part Four, we have demonstrated the applicability of these newly defined molecular
descriptors for QSPR studies. Namely, we have used them to model certain physicochemical
properties of several classes of organic compounds. The results of internal and external validations of
the obtained models have indicated that the models based on the new geometric-arithmetic indices
have high descriptive and predictive capabilities and are externally stable. Also, these results have
testified that the QSPR models based on these new topological indices in many cases outperform
models known in the literature. Therefore, it can be speculated that these new geometric-arithmetic
indices will be used in future QSPR/QSAR studies.

1. Introduction

One of the fundamental topics in all quantitative structure property/activity relationship

(QSPR/QSAR) studies is the transformation of chemical structures into molecular invariants
which, in turn, should be correlated with certain specific physicochemical properties or
biological (or toxicological/pharmacological) activities. Consequently, it is of primary
importance for any future QSPR/QSAR investigations to search for novel highly correlating
and highly discriminating molecular descriptors.
Let 𝐺 = (𝑉(𝐺), 𝐸(𝐺)) denote a molecular graph where 𝑉(𝐺) = {𝑣1 , 𝑣2 , … , 𝑣𝑛 } is the vertex
set and 𝐸(𝐺) is the edge set. The topological distance between two vertices 𝑣𝑖 , 𝑣𝑗 ∈ 𝑉(𝐺),
denoted by 𝑑𝐺 (𝑣𝑖 , 𝑣𝑗 ), is identified with the number of edges in any shortest path connecting
them. The eccentricity 𝜀𝐺 (𝑣𝑖 ) of a vertex 𝑣𝑖 ∈ 𝑉(𝐺) is the greatest topological distance
 -6-

between 𝑣𝑖 and any other vertex in 𝐺. The diameter of a molecular graph 𝐺, denoted by
𝑑𝑖𝑎𝑚(𝐺), is defined as 𝑑𝑖𝑎𝑚(𝐺) = max 𝜀𝐺 (𝑣𝑖 ). The symbol 𝑑𝑒𝑔𝐺 (𝑣𝑖 ) denotes the degree
𝑣𝑖 ∈𝑉(𝐺)

(i.e., the number of first neighbors) of the vertex 𝑣𝑖 ∈ 𝑉(𝐺). For two vertices 𝑣𝑖 , 𝑣𝑗 ∈ 𝑉(𝐺),
𝑣𝑖 𝑣𝑗 means that 𝑣𝑖 and 𝑣𝑗 are adjacent, i.e., 𝑣𝑖 𝑣𝑗 ∈ 𝐸(𝐺).

In recent years, a whole novel family of topological invariants has been introduced [11].
These new descriptors are termed as the “geometric-arithmetic indices” and their formal
definition can be expressed as follows:

√𝑓(𝑣𝑖 )𝑓(𝑣𝑗 )
𝐺𝐴𝑔𝑒𝑛𝑒𝑟𝑎𝑙 = 𝐺𝐴𝑔𝑒𝑛𝑒𝑟𝑎𝑙 (𝐺) = ∑
1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝑓(𝑣𝑖 ) + 𝑓(𝑣𝑗 ))
2

where 𝑣𝑖 , 𝑣𝑗 ∈ 𝑉(𝐺) and 𝑓(𝑣𝑖 ) is some quantity that can be uniquely connected with the
vertex 𝑣𝑖 of a molecular graph 𝐺. The first geometric-arithmetic index (𝐺𝐴1 ) was suggested
by D. Vukičević and B. Furtula by postulating 𝑓(𝑣𝑖 ) to be the degree (𝑑𝑒𝑔𝐺 (𝑣𝑖 )) of the vertex
𝑣𝑖 ∈ 𝑉(𝐺) [49]. Hence, this descriptor has the form:

√𝑑𝑒𝑔𝐺 (𝑣𝑖 )𝑑𝑒𝑔𝐺 (𝑣𝑗 )

𝐺𝐴1 (𝐺) = ∑ .
1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝑑𝑒𝑔𝐺 (𝑣𝑖 ) + 𝑑𝑒𝑔𝐺 (𝑣𝑗 ))
2

To present further geometric-arithmetic descriptors, let us recall the subsequent terminology:

for any edge 𝑣𝑖 𝑣𝑗 ∈ 𝐸(𝐺), let us define the following two quantities:

𝑛𝑣𝑖 = |{𝑥 ∈ 𝑉(𝐺)|𝑑𝐺 (𝑥, 𝑣𝑖 ) < 𝑑𝐺 (𝑥, 𝑣𝑗 )}|

and

𝑛𝑣𝑗 = |{𝑥 ∈ 𝑉(𝐺)|𝑑𝐺 (𝑥, 𝑣𝑖 ) > 𝑑𝐺 (𝑥, 𝑣𝑗 )}|.

Thus, 𝑛𝑣𝑖 is equal to the number of vertices of the molecular graph 𝐺 which are located closer
to 𝑣𝑖 ∈ 𝑉(𝐺) than to 𝑣𝑗 ∈ 𝑉(𝐺). On the other hand, the quantity 𝑛𝑣𝑗 is equal to the number of

vertices of the molecular graph 𝐺 which are located closer to 𝑣𝑗 ∈ 𝑉(𝐺) than to 𝑣𝑖 ∈ 𝑉(𝐺)
[13]. Then, the second geometric-arithmetic index, introduced by G. Fath-Tabar et al. [19],
can be expressed as follows:
 -7-

√ 𝑛 𝑣𝑖 𝑛 𝑣𝑗
𝐺𝐴2 (𝐺) = ∑ .
1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝑛 + 𝑛𝑣𝑗 )
2 𝑣𝑖

Suppose that ℎ = 𝑠𝑡 is an edge linking two vertices 𝑠, 𝑡 ∈ 𝑉(𝐺). The distance between any
vertex 𝑣𝑖 ∈ 𝑉(𝐺) and the edge ℎ in the molecular graph 𝐺 is defined as: 𝑑𝐺 (𝑣𝑖 , ℎ ) =
𝑚𝑖𝑛{𝑑𝐺 (𝑣𝑖 , 𝑠), 𝑑𝐺 (𝑣𝑖 , 𝑡)}. Then, the subsequent two quantities:

𝑚𝑣𝑖 = |{ℎ ∈ 𝐸(𝐺)|𝑑𝐺 (ℎ, 𝑣𝑖 ) < 𝑑𝐺 (ℎ, 𝑣𝑗 )}|

and

𝑚𝑣𝑗 = |{ℎ ∈ 𝐸(𝐺)|𝑑𝐺 (ℎ, 𝑣𝑖 ) > 𝑑𝐺 (ℎ, 𝑣𝑗 )}|

correspond to the number of edges of the molecular graph 𝐺 which are located closer to 𝑣𝑖 ∈
𝑉(𝐺) than to 𝑣𝑗 ∈ 𝑉(𝐺) and to the number of edges of the molecular graph 𝐺 which are
situated closer to 𝑣𝑗 ∈ 𝑉(𝐺) than to 𝑣𝑖 ∈ 𝑉(𝐺), respectively [13]. Now, the third geometric-
arithmetic index, defined by B. Zhou et. al. [50], has the form:

√ 𝑚 𝑣𝑖 𝑚 𝑣𝑗
𝐺𝐴3 (𝐺) = ∑ .
1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝑚𝑣𝑖 + 𝑚𝑣𝑗 )
2

Later on, M. Ghorbani et al. [21] suggested the fourth geometric-arithmetic index whose
formula is as follows:

√𝜀𝐺 (𝑣𝑖 )𝜀𝐺 (𝑣𝑗 )

𝐺𝐴4 (𝐺) = ∑
1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝜀𝐺 (𝑣𝑖 ) + 𝜀𝐺 (𝑣𝑗 ))
2

and A. Graovac et al. [22] considered the fifth geometric-arithmetic descriptor of the form:

√𝛿𝐺 (𝑣𝑖 )𝛿𝐺 (𝑣𝑗 )

𝐺𝐴4 (𝐺) = ∑
1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝛿 (𝑣 ) + 𝛿𝐺 (𝑣𝑗 ))
2 𝐺 𝑖

where 𝛿𝐺 (𝑣𝑖 ) = ∑𝑣𝑖 𝑢∈𝐸(𝐺) 𝑑𝑒𝑔𝐺 (𝑢).

Also, the edge and total versions of geometric-arithmetic index were introduced [33, 34].
 -8-

Some mathematical properties (e.g., lower and upper bounds, extremal graphs, inequalities,
Nordhaus-Gaddum-type results, spectral characteristic) of these indices are treated in [8, 9,
10, 11, 12, 13, 44, 45].

Also, it was demonstrated that 𝐺𝐴1 , 𝐺𝐴2 and 𝐺𝐴3 descriptors possess relatively good
descriptive as well as predictive capabilities with respect to some selected properties of
octanes and benzenoid hydrocarbons [11, 49]. The degeneracy of 𝐺𝐴1 , 𝐺𝐴2 and 𝐺𝐴3 indices
was studied in [14].

The contribution of the present report is fourfold. Firstly, we will define 9 new geometric-
arithmetic indices. Secondly, we will determine the degree of degeneracy of these newly
proposed invariants. Thirdly, we will designate minimal pairs (or subsets) of molecular graphs
having the same values of a given geometric-arithmetic index. Fourthly, we will build several
representative QSPR models to demonstrate the usefulness for chemical research of these
newly defined molecular descriptors.

To formally introduce these novel descriptors, let us recall the following terminology [24, 26,
27, 48]: the distance matrix of any molecular graph 𝐺 = (𝑉(𝐺), 𝐸(𝐺)) where |𝑉(𝐺)| = 𝑛,
denoted by 𝐷(𝐺), is a real symmetric 𝑛×𝑛 matrix whose entries [𝐷]𝑖𝑗 correspond to the
topological distance between the vertices 𝑣𝑖 , 𝑣𝑗 ∈ 𝑉(𝐺), the Harary matrix (also known as
the reciprocal distance matrix) of any molecular graph 𝐺 with 𝑛 vertices, denoted by 𝑅𝐷(𝐺),
1
is a real symmetric 𝑛×𝑛 matrix whose elements [𝑅𝐷]𝑖𝑗 are given by [𝑅𝐷]𝑖𝑗 = if 𝑣𝑖 ≠
𝑑𝐺 (𝑣𝑖 ,𝑣𝑗 )

𝑣𝑗 and [𝑅𝐷]𝑖𝑗 = 0 otherwise. On the other hand, the reverse Wiener matrix (also known as
the reverse distance matrix) of any molecular graph 𝐺 where |𝑉(𝐺)| = 𝑛, denoted by
𝑅𝑊(𝐺), is a real symmetric 𝑛×𝑛 matrix whose entries [𝑅𝑊]𝑖𝑗 are given by [𝑅𝑊]𝑖𝑗 =
𝑑𝑖𝑎𝑚(𝐺) − 𝑑𝐺 (𝑣𝑖 , 𝑣𝑗 ) if 𝑣𝑖 ≠ 𝑣𝑗 and [𝑅𝑊]𝑖𝑗 = 0 otherwise. The Randić matrix (also known
as the product connectivity matrix) of a molecular graph 𝐺 with 𝑛 vertices, denoted by 𝜒(𝐺),
is identified with a real symmetric 𝑛×𝑛 matrix whose elements [𝜒]𝑖𝑗 are given by [𝜒]𝑖𝑗 =
1
−
2
(𝑑𝑒𝑔𝐺 (𝑣𝑖 )𝑑𝑒𝑔𝐺 (𝑣𝑗 )) if 𝑣𝑖 ≠ 𝑣𝑗 and [𝜒]𝑖𝑗 = 0 otherwise.

For any molecular matrix 𝑀(𝐺) associated with a molecular graph 𝐺 = (𝑉(𝐺), 𝐸(𝐺)), the
Vertex Sum operator (also known as the Row Sum operator) for the vertex 𝑣𝑖 ∈ 𝑉(𝐺),
 -9-

denoted by 𝑉𝑆(𝑀(𝐺))𝑖 , is defined as the sum of the entries in the row 𝑖 of the graph-

theoretical matrix 𝑀(𝐺), i.e.,

𝑉𝑆(𝑀(𝐺))𝑖 = ∑[𝑀(𝐺)]𝑖𝑗 .
𝑗=1

If 𝑀(𝐺) is the distance matrix 𝐷(𝐺), then the operator 𝑉𝑆(𝑀(𝐺))𝑖 gives the distance sum of

the vertex 𝑣𝑖 ∈ 𝑉(𝐺). If 𝑀(𝐺) is the Harary matrix 𝑅𝐷(𝐺), then the operator 𝑉𝑆(𝑀(𝐺))𝑖

gives the reciprocal distance sum corresponding to the vertex 𝑣𝑖 ∈ 𝑉(𝐺) and if 𝑀(𝐺) is the
reverse Wiener matrix 𝑅𝑊(𝐺), then the operator 𝑉𝑆(𝑀(𝐺))𝑖 produces the reverse distance

sum of the vertex 𝑣𝑖 ∈ 𝑉(𝐺) [25, 48].

Consequently, it can be easily observed that for any molecular graph 𝐺 and any vertex 𝑣𝑖 ∈
𝑉(𝐺) it is possible to define the following vertex invariants: 𝑉𝑆(𝐷(𝐺))𝑖 , 𝑉𝑆(𝑅𝐷(𝐺))𝑖 ,

𝑉𝑆(𝑅𝑊(𝐺))𝑖 and 𝑉𝑆(𝜒(𝐺))𝑖 . These quantities correspond to the row sums of the distance
matrix, the reciprocal distance matrix, the reverse Wiener matrix and the product connectivity
matrix associated with the molecular graph 𝐺.

Based on the notion of the distance sum of a vertex 𝑣𝑖 ∈ 𝑉(𝐺), A. A. Dobrynin and A. A.
Kochetova introduced the so-called degree distance of 𝑣𝑖 ∈ 𝑉(𝐺) [15]. For any molecular
graph 𝐺 = (𝑉(𝐺), 𝐸(𝐺)) and any vertex 𝑣𝑖 ∈ 𝑉(𝐺), this novel vertex invariant, denoted by
𝐷′(𝑣𝑖 ), is defined as follows:

𝐷′ (𝑣𝑖 ) = 𝑑𝑒𝑔𝐺 (𝑣𝑖 )𝑉𝑆(𝐷(𝐺))𝑖 .

Other quantities which can be uniquely connected with a vertex 𝑣𝑖 ∈ 𝑉(𝐺) include the so-
called centrality measures. Such measures determine the most important elements in a given
graph 𝐺. They are mainly studied in the field of Social Network Analysis. In the present
article, we will be concerned with such centrality metrics as the eigenvector centrality, the
parameterized exponential subgraph centrality, the parameterized total subgraph
communicability, the resolvent subgraph centrality as well as with the Katz centrality.

The eigenvector centrality 𝐸𝐶𝑖 of any vertex 𝑣𝑖 ∈ 𝑉(𝐺) is identified with the 𝑖-th component
of the eigenvector associated with the largest eigenvalue of the adjacency matrix 𝐴(𝐺) of 𝐺,
i.e.,
 -10-

𝐸𝐶𝑖 = 𝒒1 (𝑖)

where 𝒒1 is the dominant eigenvector of 𝐴(𝐺) [48]. A vertex 𝑣𝑖 ∈ 𝑉(𝐺) possesses the high
value of the eigenvector centrality if it is adjacent to many other vertices or if it is linked to
other nodes that themselves have high value of this centrality measure.

Such centralities as the parameterized exponential subgraph centrality and the parameterized
total subgraph communicability are based on the notion of the parameterized matrix
exponential which is defined for any molecular graph 𝐺 by the condition

𝑒 𝛽𝐴(𝐺)

where 𝐴(𝐺) is the adjacency matrix connected with 𝐺 and 𝛽 > 0 [3, 16, 30]. The eigenvalues
of 𝑒 𝛽𝐴(𝐺) are given by 𝑒 𝛽𝜆1 , 𝑒 𝛽𝜆2 , …, 𝑒 𝛽𝜆𝑛 where 𝜆1 , 𝜆2 , …, 𝜆𝑛 are the eigenvalues of the
adjacency matrix 𝐴(𝐺). The power series expansion of 𝑒 𝛽𝐴(𝐺) is given by

2 𝑘 ∞ 𝑘
𝛽 2 (𝐴(𝐺)) 𝛽 𝑘 (𝐴(𝐺)) 𝛽 𝑘 (𝐴(𝐺))
𝑒 𝛽𝐴(𝐺) = 𝐼 + 𝛽𝐴(𝐺) + + …+ + ….= ∑ .
2! 𝑘! 𝑘!
𝑘=0

The parameterized exponential subgraph centrality of a vertex 𝑣𝑖 ∈ 𝑉(𝐺) is given by

𝑆𝐶𝑖 (𝛽) = [𝑒 𝛽𝐴(𝐺) ]𝑖𝑖 .

It is well known in the field of Graph Theory that if 𝐴(𝐺) is the adjacency matrix of a graph
𝑘
𝐺, then the entry (𝐴(𝐺))𝑖𝑗 is equal to the number of walks of length 𝑘 between two vertices

𝑣𝑖 , 𝑣𝑗 ∈ 𝑉(𝐺). Recall that a walk of length 𝑘 defined on a molecular graph 𝐺 = (𝑉(𝐺), 𝐸(𝐺))
is identified with a sequence of vertices 𝑣𝑖 , 𝑣2 , … , 𝑣𝑘+1 such that 𝑣𝑖 𝑣𝑖+1 ∈ 𝐸(𝐺) for all 1 ≤
𝑖 ≤ 𝑘. A closed walk is identified with a walk that begins and ends at the same node.
Consequently, it follows that the exponential subgraph centrality of a vertex 𝑣𝑖 ∈ 𝑉(𝐺)
(which is equal to [𝑒 𝛽𝐴(𝐺) ]𝑖𝑖 ) identifies the number of closed walks centered at 𝑣𝑖 . This
𝛽𝑘
centrality metrics weights a walk of length equal to 𝑘 by a factor 𝑘!
. Roughly speaking, the

exponential subgraph centrality estimates the number of subgraphs a node 𝑣𝑖 ∈ 𝑉(𝐺)

participates in, weighting them with respect to their size.
 -11-

On the other hand, the quantity [𝑒 𝛽𝐴(𝐺) ]𝑖𝑗 characterizes the communicability between the

vertices 𝑣𝑖 and 𝑣𝑗 in any molecular graph 𝐺. Therefore, the row sum of the matrix 𝑒 𝛽𝐴(𝐺) for a
vertex 𝑣𝑖 ∈ 𝑉(𝐺) given by
𝑛
𝑇𝐶𝑖 (𝛽) = 𝑉𝑆(𝑒 𝛽𝐴(𝐺) )𝑖 = ∑ [𝑒 𝛽𝐴(𝐺) ]𝑖𝑗
𝑗=1

identifies all walks between the vertex 𝑣𝑖 and all other vertices in the graph 𝐺 (including the
vertex 𝑣𝑖 ). In this context, the quantity 𝑇𝐶𝑖 (𝛽) is referred to as the parameterized total
subgraph communicability of the node 𝑣𝑖 ∈ 𝑉(𝐺). This centrality weights walks of length
𝛽𝑘
equal to 𝑘 by a factor 𝑘!
[3, 30].

The above-listed two centrality measures which are defined in terms of the diagonal entries or
the row sums of the parameterized exponential of the adjacency matrix of any graph 𝐺 (i.e.,
𝑒 𝛽𝐴(𝐺) ) were successfully used, for instance, in protein biochemistry (e.g., the identification
of crucial proteins in proteomic maps) [17, 18], pathophysiology (e.g., the description of
malignant tissues)[39] and neurophysiology (e.g., the characterization of healthy and stroke-
damaged brain networks) [6].

The second class of centrality measures whose formal definitions are also expressed in terms
of the matrix function 𝑓(𝐴(𝐺)) where 𝐴(𝐺) is the adjacency matrix linked with a molecular
graph 𝐺 are the so-called matrix resolvent-based centrality metrics. Recall that a matrix
−1
resolvent of 𝐴(𝐺) for a molecular graph 𝐺 = (𝑉(𝐺), 𝐸(𝐺)) is given by (𝑰 − 𝛼𝐴(𝐺)) where
1
𝑰 is the 𝑛×𝑛 identity matrix and 0 < 𝛼 < . Here, 𝜆1 denotes the spectral radius of the
𝜆1
1
adjacency matrix 𝐴(𝐺) [3, 30]. This matrix function possesses eigenvalues of the form
1−𝛼𝜆𝑖

where 𝜆𝑖 are the eigenvalues of the adjacency matrix 𝐴(𝐺). The power series expansion of
−1
(𝑰 − 𝛼𝐴(𝐺)) is given by:

∞
−1 2 𝑘 𝑘
(𝑰 − 𝛼𝐴(𝐺)) = 𝑰 + 𝛼𝐴(𝐺) + 𝛼 2 (𝐴(𝐺)) + … + 𝛼 𝑘 (𝐴(𝐺)) + … = ∑ 𝛼 𝑘 (𝐴(𝐺)) .
𝑘=0

1
The constraints imposed on the value of 𝛼 (i.e., 0 < 𝛼 < 𝜆 ) imply that the matrix 𝑰 − 𝛼𝐴(𝐺)
1

is invertible and that the above geometric series is convergent to its inverse. Such selection of
−1
𝛼 also implies that the matrix (𝑰 − 𝛼𝐴(𝐺)) is non-negative. Based on the notion of the
 -12-

−1
matrix resolvent (𝑰 − 𝛼𝐴(𝐺)) it is possible to formulate the following two definitions [3,
30]: the resolvent subgraph centrality for a vertex 𝑣𝑖 ∈ 𝑉(𝐺) of a molecular graph 𝐺 =
(𝑉(𝐺), 𝐸(𝐺)), denoted by 𝑅𝐶𝑖 (𝛼), is equal to the diagonal entries of the matrix resolvent of
the adjacency matrix 𝐴(𝐺), i.e.,

−1
𝑅𝐶𝑖 (𝛼) = [(𝑰 − 𝛼𝐴(𝐺)) ] ;
𝑖𝑖

the Katz centrality for a vertex 𝑣𝑖 ∈ 𝑉(𝐺) of a molecular graph 𝐺 = (𝑉(𝐺), 𝐸(𝐺)), denoted
by 𝐾𝑖 (𝛼), is equal to the row sums of the matrix resolvent of the adjacency matrix 𝐴(𝐺), i.e.,

𝑛
−1 −1
𝐾𝑖 (𝛼) = 𝑉𝑆 ((𝑰 − 𝛼𝐴(𝐺)) ) = ∑ [(𝑰 − 𝛼𝐴(𝐺)) ] .
𝑖 𝑖
𝑗=1

The first centrality metrics 𝑅𝐶𝑖 (𝛼) identifies the number of closed walks centered at the
vertex 𝑣𝑖 ∈ 𝑉(𝐺) whereas the second centrality metrics 𝐾𝑖 (𝛼) identifies the total number of
walks between the vertex 𝑣𝑖 ∈ 𝑉(𝐺) and all other vertices in the molecular graph 𝐺. Both
measures weights walks of length equal to 𝑘 by 𝛼 𝑘 .

Thus, we have obtained for any molecular graph 𝐺 = (𝑉(𝐺), 𝐸(𝐺)) and any node 𝑣𝑖 ∈ 𝑉(𝐺)
the following vertex invariants: 𝑉𝑆(𝐷(𝐺))𝑖 , 𝑉𝑆(𝑅𝐷(𝐺))𝑖 , 𝑉𝑆(𝑅𝑊(𝐺))𝑖 , 𝑉𝑆(𝜒(𝐺))𝑖 , 𝐷′(𝑣𝑖 ),

𝐸𝐶𝑖 , 𝑆𝐶𝑖 (𝛽), 𝑇𝐶𝑖 (𝛽), 𝑅𝐶𝑖 (𝛼) and 𝐾𝑖 (𝛼).

Based on the above considerations and on the definition of the general geometric-arithmetic
index, the following definition seems to be justified:

√𝑉𝑆(𝑀(𝐺))𝑖 𝑉𝑆(𝑀(𝐺))𝑗
𝐺𝐴6 (𝐺) = ∑
1
𝑣𝑖 𝑣𝑗∈𝐸(𝐺) (𝑉𝑆(𝑀(𝐺))𝑖 + 𝑉𝑆(𝑀(𝐺))𝑗 )
2
where 𝑀(𝐺) is any molecular matrix associated with 𝐺. Thus, in the case of the sixth
geometric-arithmetic index (𝐺𝐴6 (𝐺)), the quantity 𝑓(𝑣𝑖 ) uniquely connected with 𝑣𝑖 ∈ 𝑉(𝐺)
is identified with the row sum of 𝑀(𝐺) corresponding to the vertex 𝑣𝑖 1. Consequently, in this
paper, we will single out the following subtypes of the sixth geometric-arithmetic index:

1
Undoubtedly, if 𝑀(𝐺) is an adjacency matrix then we obtain the first geometric-arithmetic index. But when
𝑀(𝐺) is not an adjacency matrix then we will get new interesting topological indices.
 -13-

√𝑉𝑆(𝐷(𝐺))𝑖 𝑉𝑆(𝐷(𝐺))𝑗
𝐺𝐴6𝑎 (𝐺) = ∑ ,
1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝑉𝑆(𝐷(𝐺))𝑖 + 𝑉𝑆(𝐷(𝐺))𝑗 )
2

√𝑉𝑆(𝑅𝐷(𝐺))𝑖 𝑉𝑆(𝑅𝐷(𝐺))𝑗
𝐺𝐴6𝑏 (𝐺) = ∑ ,
1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝑉𝑆(𝑅𝐷(𝐺))𝑖 + 𝑉𝑆(𝑅𝐷(𝐺))𝑗 )
2

√𝑉𝑆(𝑅𝑊(𝐺))𝑖 𝑉𝑆(𝑅𝑊(𝐺))𝑗
𝐺𝐴6𝑐 (𝐺) = ∑ ,
1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝑉𝑆(𝑅𝑊(𝐺))𝑖 + 𝑉𝑆(𝑅𝑊(𝐺))𝑗 )
2

√𝑉𝑆(𝜒(𝐺))𝑖 𝑉𝑆(𝜒(𝐺))𝑗
𝐺𝐴6𝑑 (𝐺) = ∑ ,
1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝑉𝑆(𝜒(𝐺))𝑖 + 𝑉𝑆(𝜒(𝐺))𝑗 )
2

√𝑉𝑆(𝑒 𝛽𝐴(𝐺) )𝑖 𝑉𝑆(𝑒 𝛽𝐴(𝐺) )𝑗

√𝑇𝐶𝑖 (𝛽)𝑇𝐶𝑗 (𝛽)
𝐺𝐴6𝑒 (𝐺) = ∑ = ∑
1 1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝑇𝐶𝑖 (𝛽) + 𝑇𝐶𝑗 (𝛽)) 𝑣𝑖 𝑣𝑗∈𝐸(𝐺) (𝑉𝑆(𝑒 𝛽𝐴(𝐺) )𝑖 +𝑉𝑆(𝑒 𝛽𝐴(𝐺) )𝑗 )
2 2
for 𝛽 > 0,

−1 −1
√𝑉𝑆 ((𝑰 − 𝛼𝐴(𝐺)) ) 𝑉𝑆 ((𝑰 − 𝛼𝐴(𝐺)) )
√𝐾𝑖 (𝛼)𝐾𝑗 (𝛼) 𝑖 𝑗
𝐺𝐴6𝑓 (𝐺) = ∑ = ∑
1 1 −1 −1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝐾𝑖 (𝛼) + 𝐾𝑗 (𝛼)) 𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝑉𝑆 ((𝑰 − 𝛼𝐴(𝐺)) ) + 𝑉𝑆 ((𝑰 − 𝛼𝐴(𝐺)) ) )
2 2 𝑖 𝑗

1
for 0 < 𝛼 < where 𝜆1 is the spectral radius of 𝐴(𝐺).
𝜆1

In the above cases, the quantity 𝑓(𝑣𝑖 ) uniquely associated with 𝑣𝑖 ∈ 𝑉(𝐺) is identified with
the row sums (corresponding to 𝑣𝑖 ) of the following molecular matrices: the distance matrix
(𝐺𝐴6𝑎 (𝐺)), the reciprocal distance matrix (𝐺𝐴6𝑏 (𝐺)), the reverse Wiener matrix (𝐺𝐴6𝑐 (𝐺)),
the Randić matrix (𝐺𝐴6𝑑 (𝐺)), the parameterized matrix exponential of 𝐴(𝐺) (i.e., the
parameterized total subgraph communicability of the node 𝑣𝑖 ∈ 𝑉(𝐺)) (𝐺𝐴6𝑒 (𝐺)), the matrix
resolvent of 𝐴(𝐺) (i.e., the Katz centrality of the node 𝑣𝑖 ∈ 𝑉(𝐺)) (𝐺𝐴6𝑓 (𝐺)).

In the case of molecular matrices without zeros on the main diagonal, the following general
geometric-arithmetic index is proposed:

√[𝑀(𝐺)]𝑖𝑖 [𝑀(𝐺)]𝑗𝑗
𝐺𝐴7 (𝐺) = ∑ .
1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) ([𝑀(𝐺)]𝑖𝑖 + [𝑀(𝐺)]𝑗𝑗 )
2
.
 -14-

In the above expression [𝑀(𝐺)]𝑖𝑖 denotes the diagonal element corresponding to the vertex
𝑣𝑖 ∈ 𝑉(𝐺). Therefore, in this work, we will single out the following subtypes of the seventh
geometric-arithmetic index:

√[𝑒 𝛽𝐴(𝐺) ]𝑖𝑖 [𝑒 𝛽𝐴(𝐺) ]𝑗𝑗

√𝑆𝐶𝑖 (𝛽)𝑆𝐶𝑗 (𝛽)
𝐺𝐴7𝑎 (𝐺) = ∑ = ∑
1 1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝑆𝐶𝑖 (𝛽) + 𝑆𝐶𝑗 (𝛽)) 𝑣𝑖 𝑣𝑗∈𝐸(𝐺) ([𝑒 𝛽𝐴(𝐺) ]𝑖𝑖 + [𝑒 𝛽𝐴(𝐺) ]𝑗𝑗 )
2 2
for 𝛽 > 0,

−1 −1
√[(𝐼 − 𝛼𝐴(𝐺)) ] [(𝐼 − 𝛼𝐴(𝐺)) ]
√𝑅𝐶𝑖 (𝛼)𝑅𝐶𝑗 (𝛼) 𝑖𝑖 𝑗𝑗
𝐺𝐴7𝑏 (𝐺) = ∑ = ∑
1 1 −1 −1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝑅𝐶𝑖 (𝛼) + 𝑅𝐶𝑗 (𝛼)) 𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) ([(𝐼 − 𝛼𝐴(𝐺)) ] + [(𝐼 − 𝛼𝐴(𝐺)) ] )
2 2 𝑖𝑖 𝑗𝑗

1
for 0 < 𝛼 < 𝜆 where 𝜆1 is the spectral radius of 𝐴(𝐺).
1

Thus, in the cases of these subtypes, the quantity 𝑓(𝑣𝑖 ) uniquely associated with 𝑣𝑖 ∈ 𝑉(𝐺) is
identified with the parameterized exponential subgraph centrality of the node 𝑣𝑖 (𝐺𝐴7𝑎 (𝐺)) or
with the resolvent subgraph centrality of the node 𝑣𝑖 (𝐺𝐴7𝑏 (𝐺)).

Also, it seems possible to introduce the eighth geometric-arithmetic index as well as the ninth
geometric-arithmetic index. Their formal definitions are listed below:

√𝐷′(𝑣𝑖 )𝐷′(𝑣𝑗 )
𝐺𝐴8 (𝐺) = ∑ ,
1 ′
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝐷 (𝑣𝑖 ) + 𝐷′(𝑣𝑗 ))
2

√𝐸𝐶𝑖 𝐸𝐶𝑗
𝐺𝐴9 (𝐺) = ∑ ∑ .
1
𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) 𝑣𝑖 𝑣𝑗 ∈𝐸(𝐺) (𝐸𝐶𝑖 + 𝐸𝐶𝑗 )
2
In these cases, the quantity 𝑓(𝑣𝑖 ) uniquely connected with 𝑣𝑖 ∈ 𝑉(𝐺) is given by the degree
distance of the vertex 𝑣𝑖 (𝐺𝐴8 (𝐺)) or by the eigenvector centrality of the vertex 𝑣𝑖 (𝐺𝐴9 (𝐺)).

2. Datasets and computational methods

All numerical experiments were performed on a synthetic dataset of all exhaustively

generated non-isomorphic, undirected and connected graphs having up to 7 vertices with the
exception of the unique graph with |𝑉(𝐺)| = 1 and |𝐸(𝐺)| = 0. This dataset, denoted by 𝒢,
 -15-

contains 995 graphs (1 graph with |𝑉(𝐺)| = 2, 2 graphs with |𝑉(𝐺)| = 3, 6 graphs with
|𝑉(𝐺)| = 4, 21 graphs with |𝑉(𝐺)| = 5, 112 graphs with |𝑉(𝐺)| = 6 and 853 graphs with
|𝑉(𝐺)| = 7). These quantities are in agreement with the Pólya enumeration theory [40].
Graphs from the dataset 𝒢 are numbered from 1 (the graph 𝐾2 ) to 995 (graph 𝐾7 ).

In order to quantitatively assess the uniqueness (i.e., the degree of degeneracy) of a particular
molecular descriptor 𝑇𝐼, the sensitivity index 𝑆(𝑇𝐼) introduced by E. V. Konstantinova was
used [31]. This index is defined as

|𝒢| − |𝑑𝑒𝑔𝑒𝑛(𝒢)|
𝑆(𝑇𝐼) =
|𝒢|

where |𝒢| denotes the cardinality of any graph dataset 𝒢 on which 𝑇𝐼 was tested (in our case
|𝒢| = 995) and |𝑑𝑒𝑔𝑒𝑛(𝒢)| is equal to the number of degeneracies of 𝑇𝐼 within 𝒢. It is
immediately apparent that when 𝑆(𝑇𝐼) = 1, then the analyzed graph dataset 𝒢 does not
contain any pair of non-isomorphic graphs with the same values of 𝑇𝐼. Also, it can be easily
demonstrated that the sensitivity index 𝑆(𝑇𝐼) of a given topological descriptor 𝑇𝐼 is
dependent on the selected decimal places. Consequently, in this work all molecular invariants
were calculated with an accuracy of 9 decimal places.

When calculating 𝐺𝐴9 index, the eigenvector centrality is scaled so that the maximum score is
equal to 1.

The publicly available dataset of octane isomers was downloaded from the webpage
www.moleculardescriptors.eu. The dataset of 39 saturated alkanes with their experimental
boiling points (Table 12) was taken from [42]. The dataset of 29 aliphatic alcohols with their
experimental enthalpies of combustion (Table 15) was borrowed from [4, 20, 36, 38, 47]. The
dataset of 42 aliphatic alcohols with their experimental molar volumes (Table 20) was taken
from [37]. The dataset of 41 aliphatic alcohols with their experimental molar refractions
(Table 20) was also borrowed from [37]. The dataset consisting of 22 aldehydes and 24
ketones with their experimental molar refractions (Table 23) was taken from [43]. The dataset
composed of 3 aldehydes and 15 ketones with their experimental gas heat capacities (Table
26) was also borrowed from [43]. The dataset of 20 monocarboxylic acids with their
experimental enthalpies of formation and combustion (Table 31) was taken from [1, 32, 38,
46].
 -16-

In order to establish QSPR models, Linear Regression (simple and multiple) as well as Power
Regression have been used [5]. To monitor the descriptive capabilities (i.e., the goodness of
fit) of the obtained regression equations, the correlation coefficient (𝑟), the coefficient of
determination (𝑅 2 ), the standard deviation (𝑠) and the Fisher ratio (𝐹) were utilized as
statistical parameters [5, 29, 48]. As postulated by Z. Mihalić and N. Trinajstić [35], a good
QSPR model must have a value of 𝑟 > 0.99 while the values of 𝑠 depend on the property
under study. To test the predictive capabilities (i.e., the goodness of prediction) of the
obtained regression models, the leave-one-out procedure of cross-validation (𝑄 2 ) was
employed. Also, the standard deviation error in prediction (SDEP) was calculated [29, 48].
To exclude the possibility of chance correlation, the y-randomization (y-scrambling) test was
2 2
used. As suggested in [29], if 𝑅𝑦𝑟𝑎𝑛𝑑 < 0.2 and 𝑄𝑦𝑟𝑎𝑛𝑑 < 0.2, then there is no chance
2 2
correlation. Here, 𝑅𝑦𝑟𝑎𝑛𝑑 and 𝑄𝑦𝑟𝑎𝑛𝑑 stand for the basic statistics of the randomized models.
For each final regression equation, the y-randomization test was repeated 1000 times. To
verify if the obtained model has satisfactory predictive abilities with respect to external data,
we used a procedure in which the entire dataset is randomly divided into three subsets (A, B
or C) and each subset (A or B or C) is predicted by using the other two subsets (BC or AC or
AB) as the training set [28]. The quality of fit between the predicted values and the
experimental data was monitored by the values of 𝑅 2 and 𝑠.

All simulations and computations contained in the following paper were conducted in the R
programming language [7, 23, 41].

3. Degeneracy of the geometric-arithmetic indices

It is well known that when two molecular graphs are topologically identical, i.e., isomorphic,
then they also possess identical values of all graph invariants. Although, the reverse
correspondence is not universally true. This means that the identical values of any given
graph descriptors do not imply the isomorphism of the molecular graphs. Generally speaking,
any topological descriptor is said to be degenerate when there exist at least two non-
isomorphic graphs having identical values of that invariant. The uniqueness of the graph-
theoretical descriptors has been studied many times in the field of computational chemistry.
For instance, it has been observed that the level of degeneracy is high for the Wiener index
(𝑊), the Harary index (𝐻), the Hosoya index (𝑍) and the Zagreb indices (𝑍1 and 𝑍2 ), lower for
 -17-

the Randić connectivity index (𝜒) and very low for the Balaban J index [2, 14]. In order to
diminish the degree of degeneracy of first- and second-generation molecular descriptors, D.
Bonchev and N. Trinajstić developed the so-called information-theoretical indices [2].

To sum up, it can be uttered that the discriminative power is one of the fundamental
properties of each topological index. This characteristic quantitatively evaluates the capability
of molecular descriptors to distinguish non-isomorphic chemical graphs.

In the present section, our main aim is to scrutinize the extent to which the newly introduced
topological indices are degenerate as well as find the smallest pairs (or subsets) of graphs for
which the given geometric-arithmetic descriptor has the same value. The following studies are
carried out on the dataset 𝒢 of all exhaustively generated non-isomorphic, undirected and
connected graphs having from 2 to 7 vertices. Table 1 presents the values of the sensitivity
index for 𝐺𝐴1 , 𝐺𝐴4 , 𝐺𝐴6𝑎 , 𝐺𝐴6𝑏 , 𝐺𝐴6𝑐 , 𝐺𝐴6𝑑 , 𝐺𝐴6𝑒 , 𝐺𝐴6𝑓 , 𝐺𝐴7𝑎 , 𝐺𝐴7𝑏 , 𝐺𝐴8 and 𝐺𝐴9
indices, the first pairs (or subsets) of the graphs from 𝒢 having the same value of the given
geometric-arithmetic index as well as the values of that index for those minimal
indistinguishable graphs. The graphs from Table 1 are shown in Figure 1. To make our results
more illustrative, let us recall that the values of the sensitivity index 𝑆(𝑇𝐼) evaluated on the
same dataset 𝒢 for the aforementioned topological descriptors are equal to 𝑆(𝑊) = 0.011,
𝑆(𝐻) = 0.043, 𝑆(𝑍) = 0.017, 𝑆(𝑍1 ) = 0.015, 𝑆(𝑍2 ) = 0.103, 𝑆(𝜒) = 0.472 and 𝑆(𝐽) =
0.83, respectively.

Table 1. Degeneracy of twelve geometric-arithmetic indices.

The first pair (or subset) of The value of 𝐺𝐴 for
Index |𝑑𝑒𝑔𝑒𝑛(𝒢)| 𝑆(𝐺𝐴) the graphs from 𝒢 having the the graphs from the
same value of 𝐺𝐴 fourth column
𝐺𝐴1 448 0.550 43, 45 5.691642602

𝐺𝐴4 926 0.069 8, 13 4.771236166

𝐺𝐴6𝑎 173 0.826 9, 49 6

𝐺𝐴6𝑏 171 0.828 9, 49 6

𝐺𝐴6𝑐 179 0.820 28, 88, 90 7.958973274

𝐺𝐴6𝑑 70 0.930 9, 49 6

𝐺𝐴6𝑒 (β=0.005) 177 0.822 21, 46 5.999987716

𝐺𝐴6𝑒 (β=0.03) 50 0.95 9, 49 6

 -18-

𝐺𝐴6𝑒
46 0.954 9, 49 6
(0.055≤β≤0.105)

𝐺𝐴6𝑒 (0.13≤β≤9.98) 42 0.958 9, 49 6

𝐺𝐴6𝑓 (α=0.0025) 98 0.902 9, 49 6

𝐺𝐴6𝑓 (α=0.005) 59 0.941 9, 49 6

𝐺𝐴6𝑓
46-48 0.954-0.952 9, 49 6
(0.0075≤α≤0.0175)

𝐺𝐴6𝑓
(0.02≤α≤0.0325 and 44 0.956 9, 49 6
α=0.7725 )

𝐺𝐴6𝑓 (0.035≤α≤0.77
and 42 0.958 9, 49 6
0.775≤α≤0.9975)

𝐺𝐴7𝑎 (β=0.005) 983 0.012 3, 4, 5 3

𝐺𝐴7𝑎 (β=0.03) 219 0.78 43, 45, 151 5.999999747

𝐺𝐴7𝑎 (β=0.055) 49 0.951 22, 48 5.999998292

𝐺𝐴7𝑎 (β=0.08) 12 0.988 22, 48 5.999992393

𝐺𝐴7𝑎
4 0.996 9, 49 6
(0.105≤β≤3.78)

𝐺𝐴7𝑎
6 0.994 9, 49 6
(3.805≤β≤7.58)

𝐺𝐴7𝑎
8-14 0.992-0.986 9, 49 6
(7.605≤β≤9.98)

𝐺𝐴7𝑏 Many different

4-991 0.004-0.996 -
(0.0025≤α≤0.3275) pairs/subsets of graphs

𝐺𝐴7𝑏
4 0.996 9, 49 6
(0.33≤α≤0.9975)

𝐺𝐴8 169 0.830 9, 49 6

𝐺𝐴9 42 0.958 9, 49 6

From Table 1, it can be seen that the degree of degeneracy of 𝐺𝐴4 index is very high, lower
for 𝐺𝐴1 and very low for 𝐺𝐴6𝑎 , 𝐺𝐴6𝑏 , 𝐺𝐴6𝑐 , 𝐺𝐴6𝑑 , 𝐺𝐴8 and 𝐺𝐴9 indices. It was hypothesized
 -19-

that the degree of uniqueness of the geometric-arithmetic indices whose formulae include the
adjustable parameters β or α strongly depends on these parameters. Therefore, in our
computational studies, in the case of 𝐺𝐴6𝑒 and 𝐺𝐴7𝑎 indices, the adjustable parameter β had
the form of a sequence of real numbers from 0.005 to 9.98 with an increment equal to 0.025.
On the other hand, in the case of 𝐺𝐴6𝑓 and 𝐺𝐴7𝑏 indices, the adjustable parameter α had the
form of a sequence of real numbers from 0.0025 to 0.9975 with an increment equal to 0.0025.
The relationship between the adjustable parameters (β and α) and the values of 𝑆(𝑇𝐼) where
𝑇𝐼 ∈ {𝐺𝐴6𝑒 , 𝐺𝐴6𝑓 , 𝐺𝐴7𝑎 , 𝐺𝐴7𝑏 } is detailed in Figure 2. From this plot, it can be seen that the
values of 𝑆(𝐺𝐴6𝑒 ) and 𝑆(𝐺𝐴6𝑓 ) range from 0.822 to 0.958 and from 0.902 to 0.958,
respectively. On the other hand, 𝐺𝐴7𝑎 index has the very degenerate form (for β=0.005) as
well as many forms with extremely low levels of degeneracy (for 0.08≤β≤9.98) whereas
𝐺𝐴7𝑏 index has several very degenerate forms as well as many forms with extremely low
degrees of degeneracy.

Thus, in many cases we obtained the topological indices with very low level of degeneracy.

Figure 1. Graphs from Table 1.

Namely, note that the degree of degeneracy of 𝐺𝐴4 index is comparable to the degree of
degeneracy of the Wiener index (or the Harary index or two Zagreb indices) whereas the level
of degeneracy of 𝐺𝐴1 index is comparable to the level of degeneracy of the Randić index. On
the other hand, the values of the sensitivity index for 𝐺𝐴6𝑎 , 𝐺𝐴6𝑏 , 𝐺𝐴6𝑐 and 𝐺𝐴8 indices are
very close to 𝑆(𝐽) whereas 𝐺𝐴6𝑑 , 𝐺𝐴6𝑒 , 𝐺𝐴6𝑓 , 𝐺𝐴7𝑎 , 𝐺𝐴7𝑏 and 𝐺𝐴9 indices are significantly
 -20-

less degenerate than the Balaban J index or have forms with a much higher level of
uniqueness than J index.

Figure 1. Correlations between adjustable parameters (β and α) and values of sensitivity

index for four geometric-arithmetic descriptors.

4. Correlations to physical properties

It is widely recognized that topological descriptors based on molecular graphs can be easily
computed using current computer techniques. Therefore, graph-theoretical approaches are
often employed in QSAR/QSPR studies. In this section, we will demonstrate the applicability
of the newly introduced topological descriptors in modelling certain physicochemical
properties of several selected classes of organic compounds.
 -21-

4.1 The dataset of octane isomers

Saturated alkanes constitute an especially attractive family of organic compounds which are
often used as a starting point for any QSAR/QSPR investigations. One of the methodologies
employed in such studies is to select a certain class of alkanes (for instance, C8, C9 or C10
isomers) in order to obtain comparable results. In our study, we have used the dataset of
octane isomers. This reference dataset consists of 18 octane isomers and contains 16
physicochemical properties of these compounds. The dataset of octane isomers have been
used repeatedly in QSAR/QSPR research and its use for any initial assessment of modelling
properties of newly proposed topological descriptors is recommended by International
Academy of Mathematical Chemistry. Note that using the dataset of octane isomers as some
benchmark dataset it is possible to avoid the so-called size effect.

Table 2. Correlation coefficient (𝑟) between twelve geometric-arithmetic indices and nine properties of
octanes1,2.
Property
Index
BP S DENS HVAP DHVAP HFORM ACENFAC MON MV
𝐺𝐴1 0.823 0.912 -0.553 0.941 0.966 0.858 0.912 -0.777 0.538
𝐺𝐴4 0.358 0.804 -0.601 0.616 0.674 0.314 0.877 -0.729 0.617
𝐺𝐴6𝑎 0.459 0.923 -0.739 0.691 0.784 0.483 0.980 -0.905 0.752
𝐺𝐴6𝑏 0.693 0.954 -0.640 0.871 0.927 0.721 0.987 -0.922 0.639
𝐺𝐴6𝑐 0.466 0.886 -0.887 0.621 0.716 0.498 0.850 -0.813 0.880
𝐺𝐴6𝑑 0.905 0.793 -0.423 0.945 0.938 0.920 0.771 -0.565 0.396
𝐺𝐴6𝑒 0.783 0.962 -0.671 0.906 0.939 0.846 0.996 -0.943 0.673
(β) (0.005) (0.555) (0.68) (0.005) (0.055) (0.005) (1.005) (1.355) (0.805)

𝐺𝐴6𝑓 0.835 0.959 -0.660 0.937 0.955 0.889 0.996 -0.942 0.666
(α) (0.0025) (0.5875) (0.7775) (0.0025) (0.075) (0.0025) (0.805) (0.865) (0.82)

𝐺𝐴7𝑎 0.784 0.960 -0.647 0.908 0.944 0.847 0.995 -0.928 0.652
(β) (0.03) (1.93) (2.33) (0.555) (0.88) (0.005) (2.605) (3.28) (2.655)

𝐺𝐴7𝑏 0.884 0.948 -0.645 0.957 0.958 0.930 0.992 -0.938 0.653
(α) (0.005) (0.93) (0.9725) (0.005) (0.51) (0.005) (0.97) (0.9825) (0.9775)

𝐺𝐴8 0.915 0.773 -0.366 0.947 0.931 0.946 0.742 -0.526 0.338
𝐺𝐴9 0.551 0.906 -0.624 0.757 0.834 0.563 0.975 -0.922 0.636
1The values of |𝑟| greater than 0.8 are in bold.
2
In the case of 𝐺𝐴6𝑒 , 𝐺𝐴6𝑓 , 𝐺𝐴7𝑎 , 𝐺𝐴7𝑏 indices, the optimal values of the adjustable parameter (β or α)
are in parentheses below the value of the correlation coefficient.
 -22-

For the present study, we selected the following properties of octanes: the boiling point (BP),
the entropy (S), the density (DENS), the enthalpy of vaporization (HVAP), the standard
enthalpy of vaporization (DHVAP), the enthalpy of formation (HFORM), the acentric factor
(ACENFAC), the motor octane number (MON) and the molar volume (MV). The reason for
choosing these properties is that for this collection of physicochemical parameters at least one
of the tested descriptors exhibits a relatively good linear correlation(i.e., |𝑟| > 0.8). From
Table 2, it can be seen that 𝐺𝐴6𝑓 and 𝐺𝐴7𝑏 indices exhibit relatively satisfactory correlations
with seven properties of octanes, 𝐺𝐴1 , 𝐺𝐴6𝑒 and 𝐺𝐴7𝑎 indices with six properties of octanes,
𝐺𝐴6𝑏 and 𝐺𝐴6𝑐 indices with five properties of octanes, 𝐺𝐴6𝑑 , 𝐺𝐴8 and 𝐺𝐴9 indices with four
properties of octanes, 𝐺𝐴6𝑎 index with three properties of octanes as well as 𝐺𝐴4 index with
two properties of octanes. In order to further compare the descriptive and predictive abilities
of these descriptors, we constructed for each of the properties of octanes a single regression
model using only invariants with |𝑟| > 0.8. Thus, Table 3 contains the values of 𝑠 and 𝑄 2 of
five equations of the general form 𝐵𝑃 = 𝑎 + 𝑏𝐺𝐴.

Table 3. Statistical parameters of equation 𝐵𝑃 = 𝑎 + 𝑏𝐺𝐴 for five geometric-arithmetic indices1.

𝐺𝐴1 𝐺𝐴6𝑑 𝐺𝐴6𝑓 𝐺𝐴7𝑏 𝑮𝑨𝟖
(α=0.835) (α=0.005)
𝑠 3.581 2.684 3.469 6.117 2.551
𝑄2 0.539 0.713 0.588 -0.121 0.759
1
The best model is in bold.

In this case, the model based on 𝐺𝐴8 index outperforms all other models. In the case of the
model based on 𝐺𝐴8 , the improvement in the statistical deviation is equal to 28.76 %
compared to the model based on 𝐺𝐴1 index. Note that while the linear correlation between
𝐺𝐴7𝑏 index at α=0.005 and the values of BPs is is greater than 0.8, the regression model
based on this descriptor is devoid of any predictive capabilities. Table 4 presents the statistical
parameters of ten equations of the general form 𝑆 = 𝑎 + 𝑏𝐺𝐴.

Table 4. Statistical parameters of equation 𝑆 = 𝑎 + 𝑏𝐺𝐴 for ten geometric-arithmetic indices1.

𝐺𝐴1 𝐺𝐴4 𝐺𝐴6𝑎 𝐺𝐴6𝑏 𝐺𝐴6𝑐 𝑮𝑨𝟔𝒆 𝐺𝐴6𝑓 𝐺𝐴7𝑎 𝐺𝐴7𝑏 𝐺𝐴9
(β=0.555) (α=0.5875) (β=1.93) (α=0.93)
𝑠 1.915 2.771 1.792 1.389 2.160 1.266 1.327 1.307 1.476 1.967
𝑄2 0.756 0.516 0.793 0.866 0.679 0.895 0.880 0.882 0.845 0.735
1
The best model is in bold.
 -23-

In this case, the model based on 𝐺𝐴6𝑒 index at β=0.555 surpasses all other models. The
improvement in the standard deviation is equal to 33.89 % relative to the model based on the
first geometric-arithmetic index. All models exhibit good predictive abilities. In the case of
the dataset of octane isomers, the density is satisfactorily linearly correlated only with 𝐺𝐴6𝑐
index. The regression model of the form 𝐷𝐸𝑁𝑆 = 𝑎 + 𝑏𝐺𝐴6𝑐 has the values of 𝑠 and 𝑄 2
equal to 0.014 and 0.368, respectively. Table 5 includes the statistical metrics of eight
regression equations of the general form 𝐻𝑉𝐴𝑃 = 𝑎 + 𝑏𝐺𝐴. In this case, the model based on
𝐺𝐴8 index possesses the best statistical characteristics.

Table 5. Statistical parameters of equation 𝐻𝑉𝐴𝑃 = 𝑎 + 𝑏𝐺𝐴 for eight geometric-arithmetic indices1.
𝐺𝐴1 𝐺𝐴6𝑏 𝐺𝐴6𝑑 𝐺𝐴6𝑒 𝐺𝐴6𝑓 𝐺𝐴7𝑎 𝐺𝐴7𝑏 𝑮𝑨𝟖
(β=0.005) (α=0.0025) (β=0.555) (α=0.005)
𝑠 0.704 1.025 0.686 0.884 0.729 0.876 2.026 0.670
𝑄2 0.802 0.680 0.831 0.716 0.821 0.711 -0.121 0.855
1
The best model is in bold.

The improvement in the standard deviation is equal to 4.83 % relative to the model based on
𝐺𝐴1 index. The model based on 𝐺𝐴7𝑏 descriptor at α=0.005 has unsatisfactory predictive
abilities. The standard enthalpy of vaporization is satisfactorily linearly correlated with nine
geometric-arithmetic indices. The values of 𝑠 and 𝑄 2 of nine regression equations of the
general form 𝐷𝐻𝑉𝐴𝑃 = 𝑎 + 𝑏𝐺𝐴 are reported in Table 6:

Table 6. Statistical parameters of equation 𝐷𝐻𝑉𝐴𝑃 = 𝑎 + 𝑏𝐺𝐴 for nine geometric-arithmetic indices1.
𝑮𝑨𝟏 𝐺𝐴6𝑏 𝐺𝐴6𝑑 𝐺𝐴6𝑒 𝐺𝐴6𝑓 𝐺𝐴7𝑎 𝐺𝐴7𝑏 𝐺𝐴8 𝐺𝐴9
(β=0.055) (α=0.075) (β=0.88) (α=0.51)
𝑠 0.103 0.149 0.137 0.136 0.117 0.131 0.114 0.144 0.218
𝑄2 0.895 0.819 0.845 0.823 0.879 0.833 0.891 0.831 0.629
1
The best model is in bold.

In this case, the model based on 𝐺𝐴1 index has the best statistical parameters. In the case of
this model, the improvement in the standard deviation is equal to 52.75 % versus the model
based on 𝐺𝐴9 index (the worth statistical parameters). Table 7 presents the values of 𝑠 and 𝑄 2
of seven regression equations of the general form 𝐻𝐹𝑂𝑅𝑀 = 𝑎 + 𝑏𝐺𝐴.

Table 7. Statistical parameters of equation 𝐻𝐹𝑂𝑅𝑀 = 𝑎 + 𝑏𝐺𝐴 for seven geometric-arithmetic indices1.
𝐺𝐴1 𝐺𝐴6𝑑 𝐺𝐴6𝑒 𝐺𝐴6𝑓 𝐺𝐴7𝑎 𝐺𝐴7𝑏 𝑮𝑨𝟖
(β=0.005) (α=0.0025) (β=0.005) (α=0.005)
𝑠 0.661 0.5043 0.688 0.590 1.251 1.251 0.418
2
𝑄 0.685 0.806 0.653 0.752 -0.121 -0.121 0.867
1
The best model is in bold.
 -24-

In this case, the model based on 𝐺𝐴8 index is superior to all other models. The improvement
in the standard deviation is equal to 36.76 % in comparison with the model based on the 𝐺𝐴1
index. Two models (i.e., the model based on 𝐺𝐴7𝑎 index at β=0.005 and the model based on
𝐺𝐴7𝑏 index at α=0.005) lack any predictive abilities. The accentric factor is satisfactorily
linearly correlated with ten geometric-arithmetic indices.

Table 8. Statistical parameters of equation 𝐴𝐶𝐸𝑁𝐹𝐴𝐶 = 𝑎 + 𝑏𝐺𝐴 for ten geometric-arithmetic indices1.
𝐺𝐴1 𝐺𝐴4 𝐺𝐴6𝑎 𝐺𝐴6𝑏 𝐺𝐴6𝑐 𝑮𝑨𝟔𝒆 𝐺𝐴6𝑓 𝐺𝐴7𝑎 𝐺𝐴7𝑏 𝐺𝐴9
(β=1.005) (α=0.805) (β=2.605) (α=0.97)
𝑠 0.015 0.018 0.007 0.006 0.019 0.0032 0.0033 0.0037 0.0046 0.008
𝑄2 0.798 0.698 0.955 0.970 0.345 0.990 0.990 0.986 0.979 0.933
1
The best model is in bold.

The statistical parameters of ten regression equations of the general form 𝐴𝐶𝐸𝑁𝐹𝐴𝐶 = 𝑎 +
𝑏𝐺𝐴 corresponding to these descriptors are listed in Table 8. In this case, the best statistical
parameters are exhibited by the model based on 𝐺𝐴6𝑒 index at β=1.005. The improvement in
the standard deviation is equal to 78.67 %. All models possess very good predictive abilities.
The motor octane number is linearly correlated with |𝑟| > 0.8 with eight geometric-arithmetic
indices. The values of 𝑠 and 𝑄 2 of eight regression equations of the general form 𝑀𝑂𝑁 = 𝑎 +
𝑏𝐺𝐴 are presented in Table 9.

Table 9. Statistical parameters of equation 𝑀𝑂𝑁 = 𝑎 + 𝑏𝐺𝐴 for 8 geometric-arithmetic indices1.

𝐺𝐴6𝑎 𝐺𝐴6𝑏 𝐺𝐴6𝑐 𝑮𝑨𝟔𝒆 𝐺𝐴6𝑓 𝐺𝐴7𝑎 𝐺𝐴7𝑏 𝐺𝐴9
(β=1.355) (α=0.865) (β=3.28) (α=0.9825)
𝑠 10.91 9.924 14.92 8.541 8.594 9.533 8.884 9.904
𝑄2 0.751 0.803 0.556 0.846 0.846 0.810 0.837 0.802
1
The best model is in bold.

The best statistical parameters are possessed by the model based on 𝐺𝐴6𝑒 index at β=1.355.
The improvement in the value of 𝑠 is equal to 42.75 % compared to the model based on the
𝐺𝐴6𝑐 index (the worth statistical metrics). All models have good predictive capabilities. The
molar volume is satisfactorily linearly correlated with only one geometric-arithmetic index,
i.e., with 𝐺𝐴6𝑐 descriptor. The regression equation of the form 𝑀𝑉 = 𝑎 + 𝑏𝐺𝐴6𝑐 has the
values of 𝑠 and 𝑄 2 equal to 2.872 and 0.42, respectively.

From Tables 2-9, it can be inferred that in many cases the regression models based on the new
geometric-arithmetic indices perform considerably better than the regression models based on
 -25-

the first geometric-arithmetic descriptor. In the case of the dataset of octane isomers, 𝐺𝐴1
index does not exhibit any satisfactory linear correlations with such properties as the density,
the motor octane number and the molar volume. On the other hand, these properties are
linearly correlated with |𝑟| > 0.8 with 𝐺𝐴6𝑐 invariant (DENS and MV) or with 𝐺𝐴6𝑎 , 𝐺𝐴6𝑏 ,
𝐺𝐴6𝑐 , 𝐺𝐴6𝑒 (at β=1.355), 𝐺𝐴6𝑓 (at α=0.865), 𝐺𝐴7𝑎 (at β=3.28) and 𝐺𝐴7𝑏 (at β=0.9825)
indices (MON). In the case of such properties of octanes as the boiling point, the entropy, the
enthalpy of vaporization, the enthalpy of formation and the acentric factor regression models
based on one of the newly proposed geometric-arithmetic descriptors exhibit the improvement
in the standard deviation from 4.83 % (HVAP) to 78.67 % (ACENFAC) compared to models
based on the first geometric-arithmetic index.

The Pearson correlation coefficients between geometric-arithmetic indices (whose formulae

do not include the adjustable parameters α or β) defined on the dataset of octane isomers are
listed in Table 10.

The lowest linear correlation is noted between 𝐺𝐴8 and 𝐺𝐴4 indices (0.493) while the highest
linear correlation is observed between 𝐺𝐴8 and 𝐺𝐴6𝑑 indices (0.995).

Table 10. Correlation coefficients between eight geometric-arithmetic indices defined on the dataset of octane
isomers.
𝐺𝐴1 1
𝐺𝐴4 0.700 1
𝐺𝐴6𝑎 0.829 0.890 1
𝐺𝐴6𝑏 0.961 0.822 0.948 1
𝐺𝐴6𝑐 0.786 0.805 0.848 0.831 1
𝐺𝐴6𝑑 0.960 0.531 0.651 0.848 0.690 1
𝐺𝐴8 0.949 0.493 0.613 0.827 0.640 0.995 1
𝐺𝐴9 0.852 0.839 0.976 0.958 0.767 0.688 0.662 1
𝐺𝐴1 𝐺𝐴4 𝐺𝐴6𝑎 𝐺𝐴6𝑏 𝐺𝐴6𝑐 𝐺𝐴6𝑑 𝐺𝐴8 𝐺𝐴9

4.2 Correlations to the boiling points of saturated alkanes

Our initial studies have indicated that in the case of the boiling points of C2-C9 saturated
alkanes, the polynomial regression produces better models that the single regression.
Consequently, we obtained twelve equations of the general form 𝐵𝑃 = 𝑎 + 𝑏1 𝐺𝐴 + 𝑏2 𝐺𝐴2
whose statistical parameters are presented in Table 11.
 -26-

Table 11. Regression and statistical parameters of equation 𝐵𝑃 = 𝑎 + 𝑏1 𝐺𝐴 + 𝑏2 𝐺𝐴2 for twelve geometric-
arithmetic indices1.
No 𝐺𝐴 index 𝑎 𝑏1 𝑏2 𝑅2 𝐹 𝑠 𝑄2 𝑆𝐷𝐸𝑃

1b 𝐺𝐴1 78.6559 300.7592 -33.7264 0.9959 4417.41 3.2198 0.9952 3.3728

(±0.5156) (±3.2198) (±3.2198)

78.6559 299.6998 -27.5201

2b 𝐺𝐴4 0.9849 1173.381 6.2126 0.9816 6.5951
(±0.9948) (±6.2126) (±6.2126)

78.6559 299.7704 -27.2038

3b 𝐺𝐴6𝑎 0.9852 1195.204 6.1565 0.9825 6.4196
(±0.9858) (±6.1565) (±6.1565)

78.6559 299.8800 -26.0422

4b 𝐺𝐴6𝑏 0.9852 1198.667 6.1477 0.9826 6.4005
(±0.9844) (±6.1477) (±6.1477)

83.0581 249.1928 -16.1779

5b 𝐺𝐴6𝑐 0.9860 1230.23 5.0343 0.9837 5.2038
(±0.8167) (±5.0343) (±5.0343)

78.6559 301.1991 -32.0946

6b 𝑮𝑨𝟔𝒅 0.9977 7651.634 2.4486 0.9966 2.8426
(±0.3921) (±2.4486) (±2.4486)

𝐺𝐴6𝑒 78.6559 300.5076 -25.3641

7b 0.9889 1607.098 5.3194 0.9870 5.5378
(β=0.855) (±0.8518) (±5.3194) (±5.3194)

𝐺𝐴6𝑓 78.6559 300.4528 -24.6489

8b 0.9882 1504.37 5.4959 0.9862 5.7046
(α=0.9975) (±0.8801) (±5.4959) (±5.4959)

𝐺𝐴7𝑎 78.6559 301.0255 -29.0046

9b 0.9945 3232.915 3.761 0.9936 3.874
(β=1.405) (±0.6022) (±3.7610) (±3.7610)

𝐺𝐴7𝑏 78.6559 300.5712 -31.2275

10b 0.9929 2534.331 4.2446 0.9918 4.3913
(α=0.9425) (±0.6797) (±4.2446) (±4.2446)

78.6559 301.4876 -24.1181

11b 𝐺𝐴8 0.9947 3357.778 3.6907 0.9934 3.9307
(±0.5910) (±3.6907) (±3.6907)

78.6559 300.4553 -24.6235

12b 𝐺𝐴9 0.9882 1504.75 5.4952 0.9862 5.7038
(±0.8799) (±5.4952) (±5.4952)
1
The best model is in bold.

With respect to the goodness of fit, the models from Table 11 can be ordered as follows:
Eq 6b (𝐺𝐴6𝑑 ) > Eq 1b (𝐺𝐴1 ) > Eq 11b (𝐺𝐴8 ) > Eq 9b (𝐺𝐴7𝑎 (β=1.405)) > Eq 10b (𝐺𝐴7𝑏
(α=0.9425)) > Eq 7b (𝐺𝐴6𝑒 (β=0.855)) > Eq 12b (𝐺𝐴9 ) > Eq 8b (𝐺𝐴6𝑓 (α=0.9975)) > Eq 5b
(𝐺𝐴6𝑐 ) > Eq 4b (𝐺𝐴6𝑏 ) > Eq 3b (𝐺𝐴6𝑎 ) > Eq 2b (𝐺𝐴4 )

The best statistical parameters are possessed by the model based on 𝐺𝐴6𝑑 index. The
improvement in the statistical deviation is 23.95 % relative to the model based on 𝐺𝐴1 index.
The results of t-test demonstrated that all variables in this model are significant. The model of
 -27-
Eq 6b explains more than 99.76 % of the variance in the experimental values of BP for 39
alkanes. Table 12 shows the values of 𝐺𝐴6𝑑 index, the experimental boiling points of 39
saturated alkanes as well as the calculated (using Eq 6b) boiling points for this set of
compounds.

Table 12. Experimental and calculated (with Eq 6b) boiling points (BPs) of 39 saturated alkanes with values of
𝐺𝐴6𝑑 index.
BP(℃)
Subset Compound 𝐺𝐴6𝑑
Exptl Calcd
A ethane -88.630 -84.956 1
C propane -42.070 -44.794 1.886
A butane -0.500 -1.609 2.931
A 2-methylpropane -11.730 -14.856 2.598
B pentane 36.075 35.150 3.922
C 2-methylbutane 27.852 28.568 3.736
C 2, 2-dimethylpropane 9.503 8.789 3.200
C hexane 68.740 68.087 4.922
A 2-methylpentane 60.271 60.979 4.695
C 3-methylpentane 63.282 65.769 4.847
B 2, 3-dimethylbutane 57.988 57.628 4.590
A 2, 2-dimethylbutane 49.741 51.300 4.396
A heptane 98.427 96.858 5.922
B 2-methylhexane 90.052 90.819 5.699
A 3-methylhexane 91.850 93.886 5.811
B 3-ethylpentane 93.475 96.824 5.920
B 2, 4-dimethylpentane 80.500 83.796 5.450
A 2, 2-dimethylpentane 79.197 80.452 5.334
C 2, 3-dimethylpentane 89.784 90.415 5.685
A 3, 3-dimethylpentane 86.064 87.456 5.579
C 2, 2, 3-trimethylbutane 80.882 78.889 5.281
A octane 125.655 121.462 6.922
B 2-methylheptane 117.647 116.350 6.699
A 3-methylheptane 118.925 119.053 6.816
B 4-methylheptane 117.709 118.128 6.776
B 2, 5-dimethylhexane 109.103 111.087 6.479
C 3-ethylhexane 118.534 120.772 6.891
B 2, 4-dimethylhexane 109.429 113.252 6.569
B 2, 2-dimethylhexane 106.840 107.687 6.341
B 2, 3-dimethylhexane 115.607 115.229 6.652
C 3, 4-dimethylhexane 117.725 118.295 6.783
C 3, 3-dimethylhexane 111.969 112.073 6.520
A 3-ethyl-2-methylpentane 115.650 117.202 6.736
C 2, 2, 4-trimethylpentane 99.238 100.951 6.077
B 2, 3, 4-trimethylpentane 113.467 111.904 6.513
A 3-ethyl-3-methylpentane 118.259 117.339 6.742
C 2, 2, 3-trimethylpentane 109.840 108.250 6.364
B 2, 3, 3-trimethylpentane 114.760 110.455 6.453
C 2, 2, 3, 3-tetramethylbutane 106.470 98.640 5.989

The results of the y-randomization (after 1000 repetitions) produced the average value of
2 2
𝑅𝑦𝑟𝑎𝑛𝑑 equal to 0.0241 and the average value of 𝑄𝑦𝑟𝑎𝑛𝑑 equal to -0.0824. Therefore, the
 -28-

model based on 𝐺𝐴6𝑑 index does not include chance correlations. The results of external
validation of this model are presented in Table 13.

Table 13. Results of external validation of model based on 𝐺𝐴6𝑑 index.

Training set Prediction set 𝑠 𝑅2
BC A 2.9788 0.9988
AC B 2.6556 0.9917
AB C 3.1727 0.9970

Average 2.8690 0.9958

The high average value of 𝑅 2 and the relatively low average value of 𝑠 indicate that the model
of Eq 6b has good predictive abilities with respect to external data. The calculated BPs versus
the experimental data are depicted in Figure 3.

From the statistical considerations and Figure 3, we can see that the model based on 𝐺𝐴6𝑑
index is quite excellent. Note that the model of Eq 6b exhibits a lower standard deviation
than models based on 𝑋𝑢 index (𝑠 = 5.791), the Randić index χ (𝑠=7.908), the molecular
topological index (abbreviated as MTI) (𝑠=17.975) and on the Hosoya 𝑍 index (𝑠=22.924)
[35, 41].

Figure 3. Plot of calculated boiling points (BP) of 39 alkanes versus experimental data.

4.3 Correlations to the enthalpies of combustion of aliphatic alcohols

Our preliminary studies have demonstrated that the enthalpies of combustion of aliphatic
alcohols can be adequately modelled by the single regression. Thus, we obtained twelve
 -29-

equation of the general form ∆𝑐 𝐻 ° = 𝑎 + 𝑏𝐺𝐴. The statistical parameters of these models are
detailed in Table 14.

Table 14. Regression and statistical parameters of equation ∆𝑐 𝐻° = 𝑎 + 𝑏𝐺𝐴 for twelve geometric-arithmetic
Indices1.
No 𝐺𝐴 index 𝑎 𝑏 𝑅2 𝐹 𝑠 𝑄2 𝑆𝐷𝐸𝑃

-273.5408 -644.0825
1c 𝐺𝐴1 0.9991 31726.22 110.431 0.9990 114.0153
(±35.7171) (±3.6160)

-116.5654 -649.7894
2c 𝐺𝐴4 >0.9999 507545.2 27.6208 >0.9999 29.481
(±9.1124) (±0.9121)

-105.5768 -650.5732
3c 𝐺𝐴6𝑎 >0.9999 889382.5 20.8657 >0.9999 22.5035
(±6.8934) (±0.6898)

-90.6849 -651.6210
4c 𝐺𝐴6𝑏 >0.9999 1202765 17.9428 >0.9999 19.402
(±5.9390) (±0.5942)

-188.6600 -644.5356
5c 𝐺𝐴6𝑐 0.9994 41911.15 93.1386 0.9993 97.1657
(±31.9697) (±3.1483)

-237.8455 -644.7192
6c 𝐺𝐴6𝑑 0.9992 35268.79 104.7426 0.9991 108.2022
(±34.0321) (±3.4330)

𝐺𝐴6𝑒 -65.5072 -652.5242 >0.9999

7c 1564092 15.7344 >0.9999 17.0682
(β=0.13) (±5.2247) (±0.5218)

𝐺𝐴6𝑓 -68.3668 -652.3988

8c >0.9999 1570980 15.6999 >0.9999 17.0682
(α=0.275) (±5.2113) (±0.5205)

𝐺𝐴7𝑎 -66.2138 -652.4761 >0.9999

9c 1568796 15.7108 >0.9999 17.0606
(β=0.505) (±5.2164) (±0.5209)

𝑮𝑨𝟕𝒃 -69.2630 -652.2771

10c >0.9999 1578171 15.664 >0.9999 17.0784
(α=0.605) (±5.1988) (±0.5192)

-126.3731 -652.3937
11c 𝐺𝐴8 0.9999 211910.8 42.7445 0.9999 44.1835
(±14.0846) (±1.4172)

-131.5371 -655.5411
12c 𝐺𝐴9 >0.9999 357596.6 32.9057 >0.9999 34.0676
(±10.8353) (±1.0962)
1
The best model is in bold.

With respect to the goodness of fit, the models from Table 14 can be put in the following
order:

Eq 10c (𝐺𝐴7𝑏 (α=0.605)) > Eq 8c (𝐺𝐴6𝑓 (α=0.275)) > Eq 9c (𝐺𝐴7𝑎 (β=0.505)) > Eq 7c (𝐺𝐴6𝑒
(β=0.13)) > Eq 4c (𝐺𝐴6𝑏 ) > Eq 3c (𝐺𝐴6𝑎 ) > Eq 2c (𝐺𝐴4 ) > Eq 12c (𝐺𝐴9 ) > Eq 11c (𝐺𝐴8 ) >
Eq 5c (𝐺𝐴6𝑐 ) > Eq 6c (𝐺𝐴6𝑑 ) > Eq 1c (𝐺𝐴1 ).
 -30-

The best results are obtained by the model based on 𝐺𝐴7𝑏 index at α=0.605. The relationship
between the coefficient of determination (𝑅 2 ) and the adjustable parameter α (of 𝐺𝐴7𝑏
descriptor) is shown in Figure 4.

Figure 4. Plot of coefficient of determination (𝑅2 ) of equation 10c versus adjustable parameter α.

In the case of the model of Eq 10c, the improvement in the standard deviation is 85.82 %
relative to the model based on the first geometric-arithmetic index. This model explains more
than 99.99 % of the variance in the experimental data of ∆𝑐 𝐻 ° for 29 aliphatic alcohols. Table
15 contains the values of 𝐺𝐴7𝑏 index at α=0.605, the experimental enthalpies of combustion
of 29 aliphatic alcohols as well as the calculated (with Eq 10c) enthalpies of combustion for
this set of compounds.

Table 15. Experimental and calculated (with Eq 7b) the enthalpies of combustion (∆𝑐 𝐻° ) of 29 aliphatic alcohols
with values of 𝐺𝐴7𝑏 index at α=0.605.
∆𝑐 𝐻 ° (kJ/mol) 𝐺𝐴7𝑏
Subset Compound
Exptl Calcd (α=0.605)

B methanol -725.7 -721.54 1

A ethanol -1367.6 -1367.18 1.990
A 1-propanol -2019.4 -2020.98 2.992
C 2-propanol -2006.9 -2007.11 2.971
A 1-butanol -2677.4 -2674.51 3.994
B 2-butanol -2660.6 -2664.52 3.979
A 2-methyl-1-propanol -2669.6 -2664.52 3.979
C 2-methyl-2-propanol -2644 -2645.14 3.949
A 1-pentanol -3324.6 -3327.46 4.995
C 2-pentanol -3315.4 -3318.91 4.982
C 3-pentanol -3312.3 -3320.25 4.984
C 2-methyl-1-butanol -3325.9 -3320.25 4.984
B 3-methyl-1-butanol -3326.2 -3318.91 4.982
B 2-methyl-2-butanol -3303.1 -3303.43 4.958
B 3-methyl-2-butanol -3315.1 -3313.85 4.974
 -31-

B 1-hexanol -3982.6 -3980.12 5.996

C 1-heptanol -4642.52 -4632.62 6.996
A 1-octanol -5295.5 -5285.05 7.996
B 1-nonanol -5940.8 -5937.43 8.996
A 1-decanol -6599.63 -6589.78 9.997
C 1-undecanol -7253.7 -7242.12 10.997
B 1-dodecanol -7909.4 -7894.44 11.997
B 1-tridecanol -8517.8 -8546.75 12.997
C 1-tetradecanol -9167 -9199.05 13.997
B 1-pentadecanol -9817.7 -9851.35 14.997
A 1-hexadecanol -10468.9 -10503.65 15.997
A 1-octadecanol -11820 -11808 17.997
A 1-eicosanol -13130 -13112.81 19.997
C 1-docosanol -14450 -14417.38 21.997

Table 16. Results of external validation of model based on 𝐺𝐴7𝑏 index at α=0.605.
Training set Prediction set 𝑠 𝑅2
BC A 15.6270 >0.9999
AC B 18.6632 >0.9999
AB C 18.9087 >0.9999

Average 17.7330 >0.9999

2 2
The average values of 𝑅𝑦𝑟𝑎𝑛𝑑 and 𝑄𝑦𝑟𝑎𝑛𝑑 after 1000 repetitions of the y-randomization are
equal to 0.0345 and -0.1166, respectively. Thus, the model of Eq 10c does not have chance
correlations. The results of external validation of the model based on 𝐺𝐴7𝑏 index at α=0.605
are shown in Table 16. These values testify that the above model possesses satisfactory
predictive capabilities with respect to external data. The calculated enthalpies of combustion
of 29 aliphatic alcohols versus the experimental data are presented in Figure 5.

Figure 5. Plot of calculated enthalpies of combustion (∆𝑐 𝐻° ) of 29 aliphatic alcohols versus experimental data.
 -32-

From the above facts, it can be deduced that the model of Eq 10c is very good.

4.4 Correlations to the molar volumes of alcohols

From our preliminary data, it can be seen that the molar volumes of aliphatic alcohols can be
satisfactorily modelled by the single regression. Therefore, we obtained twelve linear
equations of the general form 𝑀𝑉 = 𝑎 + 𝑏𝐺𝐴 whose statistical characteristics are included in
Table 17.

Table 17. Regression and statistical parameters of equation 𝑀𝑉 = 𝑎 + 𝑏𝐺𝐴 for twelve geometric-arithmetic
indices1.
No 𝐺𝐴 index 𝑎 𝑏 𝑅2 𝐹 𝑠 𝑄2 𝑆𝐷𝐸𝑃
33.8438 16.2510
1d 𝐺𝐴1 0.9846 2564.548 4.8806 0.9821 5.1362
(±2.1054) (±0.3209)
28.1097 16.3053
2d 𝐺𝐴4 0.9984 24376.14 1.5940 0.9980 1.7020
(±0.7179) (±0.1044)
27.7540 16.3631
3d 𝐺𝐴6𝑎 0.9985 26996.75 1.5148 0.9982 1.6287
(±0.6842) (±0.0996)
27.1336 16.4368
4d 𝐺𝐴6𝑏 0.9984 25731.94 1.5516 0.9981 1.6759
(±0.7045) (±0.1025)
30.8099 15.9997
5d 𝐺𝐴6𝑐 0.9950 8016.497 2.7750 0.9943 2.9134
(±1.2233) (±0.1787)
32.4192 16.2830
6d 𝐺𝐴6𝑑 0.9853 2679.673 4.7762 0.9829 5.0249
(±2.0855) (±0.3146)
𝐺𝐴6𝑒 25.9763 16.4683
7d 0.9987 31669.18 1.3988 0.9985 1.4997
(β=0.005) (±0.6411) (±0.0925)
𝐺𝐴6𝑓 25.9755 16.4683
8d 0.9987 31674.03 1.3987 0.9985 1.4996
(α=0.0025) (±0.6411) (±0.0925)
𝑮𝑨𝟕𝒂 25.9755 16.4683
9d 0.9987 31674.39 1.3987 0.9985 1.4996
(β=0.005) (±0.6411) (±0.0925)
𝑮𝑨𝟕𝒃 25.9755 16.4683
10d 0.9987 31674.39 1.3987 0.9985 1.4996
(α=0.0025) (±0.6411) (±0.0925)
27.7760 16.6742
11d 𝐺𝐴8 0.9934 6017.976 3.2002 0.9920 3.4372
(±1.4485) (±0.2149)
28.0739 16.6984
12d 𝐺𝐴9 0.9971 13665.28 2.1276 0.9963 2.3259
(±0.9590) (±0.1428)
1
The best models are in bold.

With respect to the descriptive properties, the models from Table 17 can be ordered as
follows:

Eq 9d (𝐺𝐴7𝑎 (β=0.005)) = Eq 10d (𝐺𝐴7𝑏 (α=0.0025)) > Eq 8d (𝐺𝐴6𝑓 (α=0.0025)) > Eq 7d

(𝐺𝐴6𝑒 (β=0.005)) > Eq 3d (𝐺𝐴6𝑎 ) > Eq 4d (𝐺𝐴6𝑏 ) > Eq 2d (𝐺𝐴4 ) > Eq 12d (𝐺𝐴9 ) > Eq 5d
(𝐺𝐴6𝑐 ) > Eq 11d (𝐺𝐴8 ) > Eq 6 (𝐺𝐴6𝑑 ) > Eq 1d (𝐺𝐴1 ).
 -33-

The best statistical parameters are possessed by the models based on 𝐺𝐴7𝑎 index at β=0.005
and on 𝐺𝐴7𝑏 index at α=0.0025. The relationship between the coefficient of determination
(𝑅 2 ) and the adjustable parameters β (of 𝐺𝐴7𝑎 invariant) or α (of 𝐺𝐴7𝑏 invariant) is presented
in Figure 6.

Note that the values of 𝐺𝐴7𝑎 index at β=0.005 and 𝐺𝐴7𝑏 index at α=0.0025 are equal to the
number of edges of a molecular graph. Consequently, it can be concluded that the molar
volumes of 42 aliphatic alcohols are adequately modelled by a simple molecular descriptor,
i.e., the number of edges of the corresponding H-depleted graph. In the cases of the models of
Eqs 9d and 10d, the improvements in the standard deviation are equal to 71.34 % relative to
the model based on 𝐺𝐴1 descriptor.

Figure 6. a/ Plot of coefficient of determination (𝑅2 ) of equation 9d versus adjustable parameter β, b/ Plot of
coefficient of determination (𝑅2 ) of equation 10d versus adjustable parameter α.

These models account for more than 99.87 % of the variance in the experimental values of
MVs of 42 alcohols. Table 20 presents the values of 𝐺𝐴7𝑎 index at β=0.005 (or 𝐺𝐴7𝑏 index at
α=0.0025), the experimental molar volumes of 42 aliphatic alcohols as well as the calculated
(with Eqs 9d or 10d) molar volumes for this set of compounds.

2 2
After 1000 repetitions, the y-scrambling produced the average values of 𝑅𝑦𝑟𝑎𝑛𝑑 and 𝑄𝑦𝑟𝑎𝑛𝑑
equal to 0.0246 and -0.0762, respectively. Consequently, the above models do not have
chance correlations. Table 18 presents the results of external validation of the models of Eqs
9d and 10d.
 -34-

Table 18. Results of external validation of the model based on 𝐺𝐴7𝑎 index at β=0.005 (or on 𝐺𝐴7𝑏 index at
α=0.0025).
Training set Prediction set 𝑠 𝑅2
BC A 1.2300 0.9993
AC B 1.5429 0.9990
AB C 1.9416 0.9986

Average 1.5715 0.9990

The high average value of 𝑅 2 and the low average value of 𝑠 indicate that these models
exhibit very good predictive abilities for external data. The plot of the calculated MVs of 42
aliphatic alcohols versus the experimental data is shown in Figure 7. It can be observed that
the calculated values of MVs agree very well with the experimental data. Judging from the
statistical parameters and plot in Figure 7, it can be uttered that the regression models based
on 𝐺𝐴7𝑎 index at β=0.005 or on 𝐺𝐴7𝑏 index at α=0.0025 represent excellent QSPR models.

Figure 7. Plot of calculated molar volumes (MV) of 42 aliphatic alcohols versus experimental data.

For this same dataset, L. Mu et al. obtained the three-parameter regression model (with two
0 1
edge connectivity indices, ie., 𝐹, 𝐹 and the alcohol-type parameter δ) with a slightly
higher standard deviation (𝑠=1.504) [37].

4.5 Correlations to the molar refractions of alcohols

The molar refraction (MR) is a measure of the total polarizability of molecules. This property
is a particularly important physical characteristic in chemistry, biochemistry and
pharmaceutical sciences. Our initial results have indicated that in the case of 41 aliphatic
alcohols this property can be adequately described by the single regression. Therefore, we
obtained twelve linear regression equations of the general form 𝑀𝑅 = 𝑎 + 𝑏𝐺𝐴. The
statistical parameters exhibited by these models are listed in Table 19.
 -35-

Table 19. Regression and statistical parameters of equation 𝑀𝑅 = 𝑎 + 𝑏𝐺𝐴 for twelve geometric-arithmetic
indices1.
No 𝐺𝐴 index 𝑎 𝑏 𝑅2 𝐹 𝑠 𝑄2 𝑆𝐷𝐸𝑃
5.7455 4.5620
1e 𝐺𝐴1 0.9888 3437.633 1.1745 0.9871 1.2263
(±0.5141) (±0.0778)
4.2815 4.5599
2e 𝐺𝐴4 0.9996 104834.8 0.2138 0.9996 0.2244
(±0.0974) (±0.0141)
4.1675 4.5774
3e 𝐺𝐴6𝑎 0.9997 135036.8 0.1884 0.9997 0.1984
(±0.0861) (±0.0125)
3.9960 4.5979
4e 𝐺𝐴6𝑏 0.9997 130301.7 0.1918 0.9996 0.2041
(±0.0881) (±0.0127)
5.0053 4.4782
5e 𝐺𝐴6𝑐 0.9971 13564.76 0.5937 0.9967 0.6201
(±0.2649) (±0.0385)
5.3520 4.5705
6e 𝐺𝐴6𝑑 0.9898 3769.446 1.1221 0.9882 1.1728
(±0.4969) (±0.0744)
𝐺𝐴6𝑒 3.7479 4.6044
7e 0.9997 154562.6 0.1761 0.9997 0.1849
(β=0.105) (±0.0815) (±0.0117)
𝑮𝑨𝟔𝒇 3.7732 4.6031
8e 0.9997 154622.6 0.1761 0.9997 0.1851
(α=0.235) (±0.0814) (±0.0117)
𝐺𝐴7𝑎 3.7516 4.6043
9e 0.9997 154037 0.1764 0.9997 0.1854
(β=0.455) (±0.0816) (±0.0117)
𝐺𝐴7𝑏 3.7963 4.5997
10e 0.9997 152840.8 0.1771 0.9997 0.1858
(α=0.5875) (±0.0818) (±0.0118)
4.1199 4.6715
11e 𝐺𝐴8 0.9964 10661.6 0.6694 0.9956 0.7138
(±0.3068) (±0.0452)
4.2377 4.6736
12e 𝐺𝐴9 0.9987 31131.43 0.3922 0.9985 0.4221
(±0.1789) (±0.0265)
1
The best model is in bold.
With regard to the goodness of fit, the models from Table 19 can be ordered as follows:

Eq 8e (𝐺𝐴6𝑓 (α=0.235)) > Eq 7e (𝐺𝐴6𝑒 (β=0.105)) > Eq 9e (𝐺𝐴7𝑎 (β=0.455)) > Eq 10e (𝐺𝐴7𝑏
(α=0.5875)) > Eq 3e (𝐺𝐴6𝑎 ) > Eq 4e (𝐺𝐴6𝑏 ) > Eq 2e (𝐺𝐴4 ) > Eq 12e (𝐺𝐴9 ) > Eq 5e (𝐺𝐴6𝑐 ) >
Eq 11e (𝐺𝐴8 ) > Eq 6e (𝐺𝐴6𝑑 ) > Eq 1e (𝐺𝐴1 ).

Figure 8. Plot of coefficient of determination (𝑅2 ) of equation 8e versus adjustable parameter α.

 -36-

The model of Eq 8e shows the best statistical parameters. The relationship between the
coefficient of determination (𝑅 2 ) and the adjustable parameter α (of 𝐺𝐴6𝑓 invariant) is plotted
in Figure 8. In the case of this model, the improvement in the statistical deviation is equal to
85.01 % relative to the model of Eq 1e. The model based on 𝐺𝐴6𝑓 index at α=0.235 is
responsible for more than 99.97 % of the variance in the experimental MRs of 41 aliphatic
alcohols. The values of 𝐺𝐴6𝑓 index at α=0.235, the experimental values of MRs as well as the
calculated (with Eq 8e) values of MRs for this set of compounds are presented in Table 20.
Table 20. Experimental and calculated (with Eq 9d/10d (MV) or Eq 8e (MR)) the molar volumes (MVs) and the
molar refractions (MRs) of aliphatic alcohols with values of 𝐺𝐴7𝑎 index at β=0.005 (or 𝐺𝐴7𝑏 index at α=0.0025)
and 𝐺𝐴6𝑓 index at α=0.235.
MV(cm3/mol) 𝐺𝐴7𝑎 MR(cm3/mol)
(β=0.005) 𝐺𝐴6𝑓
Subset Compound
Exptl Calcld 𝐺𝐴7𝑏 Exptl Calcd (α=0.235)
(α=0.0025)
A ethanol 58.368 58.912 2 12.927 12.959 1.996
C 1-propanol 74.798 75.380 3 17.565 17.561 2.995
B 2-propanol 76.561 75.380 3 17.613 17.504 2.983
C 1-butanol 91.529 91.849 4 22.145 22.166 3.996
B 2-methyl-1-propanol 92.338 91.849 4 22.182 22.117 3.985
C 2-butanol 91.903 91.849 4 22.144 22.117 3.985
B 2-methyl-2-propanol 94.216 91.849 4 22.033 22.014 3.963
A 1-pentanol 108.160 108.317 5 26.798 26.771 4.996
B 3-methyl-1-butanol 108.559 108.317 5 26.770 26.725 4.986
B 2-pentanol 108.962 108.317 5 26.724 26.725 4.986
B 2-methyl-1-butanol 108.027 108.317 5 26.753 26.730 4.987
A 3-pentanol 107.265 108.317 5 26.565 26.730 4.987
A 3-methyl-2-butanol 107.631 108.317 5 26.638 26.687 4.978
B 2-methyl-2-butanol 108.962 108.317 5 26.718 26.633 4.966
B 2, 2-dimethyl-1-propanol 108.559 108.317 5 - - -
C 1-hexanol 125.590 124.785 6 31.636 31.375 5.996
A 2-methyl-1-pentanol 123.795 124.785 6 31.262 31.337 5.988
C 2-ethyl-1-butanol 122.401 124.785 6 31.130 31.344 5.990
C 4-methyl-2-pentanol 126.774 124.785 6 31.497 31.292 5.978
C 2, 3-dimethyl-2-butanol 124.065 124.785 6 31.239 31.213 5.961
B 3, 3-dimethyl-1-butanol 124.005 124.785 6 31.224 31.242 5.968
B 3, 3-dimethyl-2-butanol 124.838 124.785 6 31.268 31.213 5.961
A 3-hexanol 124.716 124.785 6 31.297 31.337 5.988
A 3-methyl-3-pentanol 123.391 124.785 6 31.134 31.251 5.969
A 1-heptanol 141.345 141.253 7 36.015 35.978 6.996
C 2-heptanol 142.176 141.253 7 36.077 35.934 6.987
B 3-heptanol 141.535 141.253 7 35.981 35.942 6.988
A 4-heptanol 142.002 141.253 7 35.928 35.944 6.989
A 2, 4-dimethyl-3-pentanol 140.101 141.253 7 35.794 35.875 6.974
C 1-octanol 157.473 157.722 8 40.679 40.582 7.996
A 2-octanol 158.720 157.722 8 40.668 40.538 7.987
C 4-octanol 158.972 157.722 8 40.649 40.548 7.989
A 2-ethyl-1-hexanol 156.357 157.722 8 40.514 40.554 7.990
C 2, 2, 4-trimethyl-1-pentanol 155.221 157.722 8 40.097 40.432 7.964
A 3, 5-dimethyl-1-hexanol 156.960 157.722 8 40.135 40.510 7.981
B 1-nonanol 174.417 174.190 9 45.266 45.185 8.997
B 2, 6-dimethyl-4-heptanol 177.638 174.190 9 45.244 45.078 8.973
 -37-

A 5-nonanol 172.642 174.190 9 44.589 45.152 8.989

B 1-decanol 190.252 190.658 10 49.734 49.788 9.997
C 1-undecanol 207.652 207.127 11 54.640 54.392 10.997
C 2, 6, 8-trimethyl-4-nonanol 227.438 223.595 12 59.289 58.862 11.968
C 1-tridecanol 236.965 240.063 13 63.375 63.598 11.997

2 2
In the case of the model of Eq 8e, the average values of 𝑅𝑦𝑟𝑎𝑛𝑑 and 𝑄𝑦𝑟𝑎𝑛𝑑 after 1000
repetitions of the y-scrambling are equal to 0.0253 and -0.0792, respectively.
Hence, the model of Eq 8e does not contain chance correlations. The results of external
validation of the model based on 𝐺𝐴6𝑓 index at α=0.235 are shown in Table 21.

Table 21. Results of external validation of model based on 𝐺𝐴6𝑓 index at α=0.235.
Training set Prediction set 𝑠 𝑅2
BC A 0.2452 0.9996
AC B 0.0992 >0.9999
AB C 0.2588 0.9998

Average 0.2011 0.9998

Figure 9. Plot of calculated molar refractions (MR) of 41 aliphatic alcohols versus experimental data.

The values of 𝑅 2 and 𝑠 suggest that the above model exhibits satisfactory predictive
capabilities with respect to external data. From the plot in Figure 9, it can be seen that the
calculated values of MRs of 41 alcohols are very close to the experimental data. To sum up,
the model based on 𝐺𝐴6𝑓 index at α=0.235 can be referred as very good. For this same
0 1
dataset, L. Mu et al. obtained the three-parameter model (with 𝐹, 𝐹 and δ as variables)
with a higher standard deviation (𝑠=0.446) [37].
 -38-

4.6 Correlations to the molar refractions of aldehydes and ketones

Also, the molar refractions of the set of compounds composed of 22 aldehydes and 24 ketones
are properly modelled by the single regression. So, twelve linear regression equations of the
form 𝑀𝑅 = 𝑎 + 𝑏𝐺𝐴 with their statistical parameters are presented in Table 22.

Table 22. Regression and statistical parameters of equation 𝑀𝑅 = 𝑎 + 𝑏𝐺𝐴 for twelve geometric-arithmetic
indices1.
No 𝐺𝐴 index 𝑎 𝑏 𝑅2 𝐹 𝑠 𝑄2 𝑆𝐷𝐸𝑃

3.7920 4.6280
1f 𝐺𝐴1 0.9946 8072.622 0.8642 0.9942 0.8780
(±0.3721) (±0.0515)

2.7239 4.5891
2f 𝐺𝐴4 0.9997 131361.6 0.2148 0.9996 0.2222
(±0.0950) (±0.0127)

2.6377 4.6017
3f 𝐺𝐴6𝑎 0.9997 165811.4 0.1912 0.9997 0.1962
(±0.0848) (±0.0113)

2.4783 4.6217
4f 𝐺𝐴6𝑏 0.9998 179960.2 0.1835 0.9997 0.1877
(±0.0817) (±0.0109)

3.0954 4.5479
5f 𝐺𝐴6𝑐 0.9991 51605.61 0.3426 0.9990 0.3569
(±0.1501) (±0.0200)

3.4270 4.6299
6f 𝐺𝐴6𝑑 0.9949 8559.956 0.8394 0.9945 0.8515
(±0.3650) (±0.0500)

𝐺𝐴6𝑒 2.2712 4.6272

7f 0.9998 182187.6 0.1824 0.9997 0.1869
(β=0.13) (±0.0817) (±0.0108)

𝑮𝑨𝟔𝒇 2.3176 4.6262

8f 0.9998 186003.6 0.1805 0.9997 0.1849
(α=0.315) (±0.0808) (±0.0107)

𝐺𝐴7𝑎 2.2635 4.6272

9f 0.9998 180232.9 0.1834 0.9997 0.1879
(β=0.48) (±0.0822) (±0.0109)

𝐺𝐴7𝑏 2.3344 4.6226

10f 0.9998 182320.5 0.1823 0.9997 0.1869
(α=0.64) (±0.0815) (±0.0108)

2.4545 4.7090
11f 𝐺𝐴8 0.9980 22419.53 0.5195 0.9979 0.5264
(±0.2317) (±0.0315)

2.6683 4.7015
12f 𝐺𝐴9 0.9991 51000.95 0.3446 0.9991 0.3515
(±0.1527) (±0.0208)
1
The best model is in bold.

With respect to the decreasing goodness of fit, the models from Table 22 can be put in the
following order:
 -39-

Eq 8f (𝐺𝐴6𝑓 (α=0.315)) > Eq 10f (𝐺𝐴7𝑏 (α=0.64)) > Eq 7f (𝐺𝐴6𝑒 (β=0.13)) > Eq 9f (𝐺𝐴7𝑎
(β=0.48)) > Eq 4f (𝐺𝐴6𝑏 ) > Eq 3f (𝐺𝐴6𝑎 ) > Eq 2f (𝐺𝐴4 ) > Eq 5f (𝐺𝐴6𝑐 ) > Eq 12f (𝐺𝐴9 ) > Eq
11f (𝐺𝐴8 ) > Eq 6f (𝐺𝐴6𝑑 ) > Eq 1 (𝐺𝐴1 ).

Also, in this case, the best parameters are possessed by the model based on 𝐺𝐴6𝑓 index at
α=0.315. Figure 10 presents the mutual relation between the coefficient of determination (𝑅 2 )
and the adjustable parameter α (of 𝐺𝐴6𝑓 descriptor).

Figure 10. Plot of coefficient of determination (𝑅2 ) of equation 8f versus adjustable parameter α.

In the case of the model of Eq 8f, the improvement in the standard deviation is equal to 79.11
% compared to the model based on 𝐺𝐴1 index. This model elucidates more than 99.97 % of
variance in the experimental data of MRs for this set of compounds. Table 23 presents the
values of 𝐺𝐴6𝑓 index at α=0.315, the experimental values of the molar refractions of 22
aldehydes and 24 ketones as well as the calculated (with Eq 8f) values of MRs for this set of
compounds.

Table 23. Experimental and calculated (with Eq 8f ) the molar refractions of 22 aldehydes and 24 ketones with
values of 𝐺𝐴6𝑓 index at α=0.315.
MR(cm3/mol) 𝐺𝐴6𝑓
Subset Compound
Exptl Calcd (α=0.315)

C acetaldehyde 11.5829 11.5377 1.993

B propionaldehyde 16.1632 16.1597 2.992
A butyl aldehyde 20.8011 20.7878 3.993
A 2-methyl propanal 20.8219 20.7092 3.976
B pentaldehyde 25.4983 25.4161 4.993
B 2-methyl butanal 25.3943 25.3473 4.978
A 3-methyl butanal 25.5327 25.3431 4.977
B hexanal 30.0928 30.0438 5.993
 -40-

B 2-methylpentanal 29.8497 29.9803 5.980

B 2-ethylbutanal 29.9981 29.9870 5.981
A 2, 3-dimethylbutanal 30.0640 29.9172 5.966
B heptanal 34.7004 34.6710 6.994
A 2, 2-dimethylpentanal 34.7537 34.4780 6.952
C octanal 39.4396 39.2979 7.994
C 2-ethylhexanal 39.2395 39.2480 7.983
C 2-ethyl-3-methylpentanal 38.9423 39.1985 7.972
C nonanal 44.2669 43.9246 8.994
A 3, 5, 5-trimethylhexanal 43.9887 43.6785 8.941
B decanal 48.6737 48.5512 9.994
A 2-methyldecanal 53.0003 53.1163 10.981
C dodecanal 58.0913 57.8041 11.994
C 2-methylundecanal 57.9284 57.7426 11.981
B acetone 16.2963 16.0722 2.973
C 2-butanone 20.6039 20.7092 3.976
A 2-pentanone 25.2926 25.3431 4.977
B 3-pentanone 25.2487 25.3473 4.978
A 3-methyl-2-butanone 25.2603 25.2765 4.963
B 2-hexanone 29.9308 29.9728 5.978
A 3-hexanone 29.7251 29.9803 5.980
C 3-methyl-2-pentanone 29.9453 29.9172 5.966
C 4-methyl-2-pentanone 29.9877 29.9100 5.964
A 3, 3-dimethyl-2-butanone 29.6748 29.7793 5.936
B 2-heptanone 34.5463 34.6008 6.978
C 3-heptanone 34.4230 34.6092 6.980
B 4-heptanone 34.3083 34.6126 6.981
A 5-methyl-2-hexanone 34.5773 34.5360 6.964
A 2-octanone 39.1959 39.2280 7.979
C 4-octanone 39.0616 39.2410 7.981
C 6-methyl-3-heptanone 38.9478 39.1724 7.967
A 2-nonanone 43.3542 43.8549 8.979
C 5-nonanone 43.8710 43.8692 8.982
A 2, 6-dimethyl-4-heptanone 43.8902 43.7491 8.956
B 2-decanone 48.5304 48.4816 9.979
C 2-undecanone 52.7129 53.1082 10.979
C 6-undecanone 53.2109 53.1230 10.982
B 2-methyl-4-undecanone 57.7027 57.6878 11.969

In the case of the model of Eq 8f, the y-randomization (after 1000 repetitions) gave the
2 2
average value of 𝑅𝑦𝑟𝑎𝑛𝑑 equal to 0.0229 and the average value of 𝑄𝑦𝑟𝑎𝑛𝑑 equal to -0.0703.
The results of external validation of this model are presented in Table 24:

Table 24. Results of external validation of model based on 𝐺𝐴6𝑓 index at α=0.315.
Training set Prediction set 𝑠 𝑅2
BC A 0.2189 0.9995
AC B 0.1257 0.9999
AB C 0.2158 0.9998

Average 0.1868 0.9997

 -41-

The values from Table 24 indicate that the model based on 𝐺𝐴6𝑓 index at α=0.315 has good
predictive ability for external data. The plot of the calculated values of MRs of 22 aldehydes
and 24 ketones versus the experimental data is depicted in Figure 11. This figure as well as all
statistical conditions support the view that the model of Eq 8f can be considered as excellent.
For this same dataset, B. Ren obtained the six-parameter model (with the modified 𝑋𝑢 index,
i.e., 𝑋𝑢𝑢𝑚 and five atom-type-based AI topological indices) with a slightly lower standard
deviation (𝑠=0.1598) [43].

Figure 11. Plot of calculated molar refractions of 22 aldehydes and 24 ketones versus experimental data.

4.7 Correlations to the gas heat capacities of aldehydes and ketones

In the case of the gas heat capacities of the set of compounds composed of 3 aldehydes and 15
ketones, our preliminary studies have demonstrated that the power regression produces better
models than any linear regression. Thus, we obtained twelve nonlinear equations of the
general form 𝐶𝑝𝐺 = 𝑐𝐺𝐴𝑡 . The statistical metrics of these models are presented in Table 25.

Table 25. Regression and statistical parameters of equation 𝐶𝑝𝐺 = 𝑐𝐺𝐴𝑡 for twelve geometric-arithmetic indices1.
No 𝐺𝐴 index 𝑐 𝑡 𝑅2 𝐹 𝑠 𝑄2 𝑆𝐷𝐸𝑃

𝑒𝑥𝑝(3.6482) 0.8214
1g 𝐺𝐴1 0.9923 2051.123 0.0236 0.9903 0.0249
(±0.0302) (±0.0181)

𝑒𝑥𝑝(3.5678) 0.8376
2g 𝐺𝐴4 0.9866 1179.175 0.0310 0.9837 0.0323
(±0.0421) (±0.0244)

𝑒𝑥𝑝(3.5527) 0.8465
3g 𝐺𝐴6𝑎 0.9869 1204.717 0.0306 0.9833 0.0326
(±0.0421) (±0.0244)
 -42-

𝑒𝑥𝑝(3.5378) 0.8540
4g 𝐺𝐴6𝑏 0.9862 1141.826 0.0315 0.9823 0.0336
(±0.0437) (±0.0253)

𝑒𝑥𝑝(3.6362) 0.8033
5g 𝐺𝐴6𝑐 0.9897 1536.597 0.0272 0.9867 0.0290
(±0.0352) (±0.0205)

𝑒𝑥𝑝(3.6345) 0.8221
6g 𝐺𝐴6𝑑 0.9901 1603.255 0.0266 0.9877 0.0280
(±0.0345) (±0.0205)

𝐺𝐴6𝑒 𝑒𝑥𝑝(3.5484) 0.8588

7g 0.9907 1702.669 0.0258 0.9878 0.0279
(β=1.505) (±0.0355) (±0.0208)

𝐺𝐴6𝑓 𝑒𝑥𝑝(3.5420) 0.8645

8g 0.9903 1637.899 0.0263 0.9874 0.0284
(α=0.9975) (±0.0364) (±0.0214)

𝐺𝐴7𝑎 𝑒𝑥𝑝(3.6338) 0.8617

9g 0.9945 2873.417 0.0199 0.9924 0.0220
(β=31.205) (±0.0258) (±0.0161)

𝑮𝑨𝟕𝒃 𝒆𝒙𝒑(3.6410) 0.8291

10g 0.9945 2915.997 0.0198 0.9927 0.0216
(α=0.97) (±0.0255) (±0.0154)

𝑒𝑥𝑝(3.5387) 0.8643
11g 𝐺𝐴8 0.9892 1465.186 0.0278 0.9863 0.0295
(±0.0385) (±0.0226)

𝑒𝑥𝑝(3.5418) 0.8647
12g 𝐺𝐴9 0.9903 1639.367 0.0263 0.9874 0.0284
(±0.0364) (±0.0214)
1
The best model is in bold.

With respect to the descriptive properties, the models from Table 25 can be ordered as
follows:

Eq 10g (𝐺𝐴7𝑏 (α=0.97)) > Eq 9g (𝐺𝐴7𝑎 (β=31.205)) > Eq 1g (𝐺𝐴1 ) > Eq 7g (𝐺𝐴6𝑒 (β=1.505))
> Eq 12g (𝐺𝐴9 ) > Eq 8g (𝐺𝐴6𝑓 (α=0.9975)) > Eq 6g (𝐺𝐴6𝑑 ) > Eq 5g (𝐺𝐴6𝑐 ) > Eq 11g (𝐺𝐴8 )
> Eq 3g (𝐺𝐴6𝑎 ) > Eq 2g (𝐺𝐴4 ) > Eq 4g (𝐺𝐴6𝑏 ).
The model based on 𝐺𝐴7𝑏 index at α=0.97 outperforms all other models. The relationship
between the coefficient of determination (𝑅 2 ) and the adjustable parameter α (of 𝐺𝐴7𝑏
descriptor) is presented in Figure 12.
 -43-

Figure 12. Plot of coefficient of determination (𝑅2 ) of equation 10g versus adjustable parameter α.

In the case of the model based on 𝐺𝐴7𝑏 index at α=0.97, the improvement in the standard
deviation is equal to 16.10 % versus the model of Eq 1g. More than 99.45 % of the variance
in the experimental data of the gas heat capacities of the considered set of compounds is
explained by this model. The values of 𝐺𝐴7𝑏 index at α=0.97, the experimental gas heat
capacities of the set of 3 aldehydes and 15 ketones as well as the calculated (with Eq 10g)
values of 𝐶𝑝𝐺 for this set of compounds are listed in Table 26.

Table 26. Experimental and calculated (with Eq 10g) gas heat capacities (𝐶𝑝𝐺 ) of 3 aldehydes and 15 ketones
with values of 𝐺𝐴7𝑏 index at α=0.97.
𝐶𝑝𝐺 (J/molK) 𝐺𝐴7𝑏
Subset Compound
Exptl Calcd (α=0.97)

B propanal 90.03 90.64 2.842

A pentanal 144.07 140.21 4.809
A 2, 2-dimethylpropanal 132.42 127.62 4.293
B acetone 83.99 86.04 2.668
C 2-butanone 110.02 110.64 3.614
A 2-pentanone 136.23 135.57 4.618
C 3-pentanone 133.54 134.35 4.568
B 3-methyl-2-butanone 131.09 130.98 4.430
C 2-hexanone 161.50 159.99 5.639
A 3-hexanone 157.82 158.61 5.581
B 4-methyl-2-pentanone 155.68 156.34 5.484
C 3, 3-dimethyl-2-butanone 149.64 148.12 5.139
A 2-heptanone 189.55 183.69 6.662
C 4-heptanone 180.63 182.21 6.597
B 2-methyl-3-hexanone 173.25 178.28 6.426
B 2, 4-dimethyl-3-pentanone 171.98 174.70 6.270
A 2-octanone 209.54 206.70 7.680
C 5-nonanone 221.35 228.05 8.648
 -44-

Table 27. Results of external validation of model based on 𝐺𝐴7𝑏 index at α=0.97.
Training set Prediction set 𝑠 𝑅2
BC A 0.0388 0.9925
AC B 0.0333 0.9987
AB C 0.0222 0.9975

Average 0.0314 0.9962

In the case of the model based on 𝐺𝐴7𝑏 index at α=0.97, the y-scrambling (after 1000
2
repetitions) produced the average value of 𝑅𝑦𝑟𝑎𝑛𝑑 equal to 0.0562 and the average value of
2
𝑄𝑦𝑟𝑎𝑛𝑑 equal to -0.2065. Thus, this model does not possess chance correlations.

Table 27 presents the results of external validation of the model based on 𝐺𝐴7𝑏 index at
α=0.97. The high average value of 𝑅 2 and the low value of 𝑠 testify that this model can be
reckoned as having very good predictive abilities with regard to external data. From the plot
in Figure 13, it can be deduced that there exists agreement between the calculated (with Eq
10g) gas heat capacities and the experimental data. All aforementioned facts indicate that the
model based on 𝐺𝐴7𝑏 index at α=0.97 is of high quality.

Figure 13. Plot of calculated gas heat capacities (𝐶𝑝𝐺 ) of 3 aldehydes and 15 ketones versus experimental data.

For this same dataset, B. Ren obtained the two-parameter model (with 𝑋𝑢𝑢𝑚 index and one AI
index) with a higher standard deviation (𝑠=2.48) [43].
 -45-

4.8 Correlations to the enthalpies of formation of monocarboxylic acids

Our introductory findings suggest that the enthalpies of formation of 20 monocarboxylic acids
can be satisfactorily modelled by the single regression. Hence, we obtained twelve linear
regression equations of the general form ∆𝑓 𝐻 ° = 𝑎 + 𝑏𝐺𝐴 whose statistical characteristics are
presented in Table 28. With regard to the decreasing goodness of fit, the models from Table
28 can be arranged as follows:
Eq 11h (𝐺𝐴8 ) > Eq 7h (𝐺𝐴6𝑒 (β=9.98)) > Eq 9h (𝐺𝐴7𝑎 (β=0.005)) = Eq 10h (𝐺𝐴7𝑏
(α=0.0025)) > Eq 8h (𝐺𝐴6𝑓 (α=0.0025)) > Eq 12h (𝐺𝐴9 ) > Eq 4h (𝐺𝐴6𝑏 ) > Eq 3h (𝐺𝐴6𝑎 ) >
Eq 2h (𝐺𝐴4 ) > Eq 6h (𝐺𝐴6𝑑 ) > Eq 1h (𝐺𝐴1 ) > Eq 5h (𝐺𝐴6𝑐 ).
Table 28. Regression and statistical parameters of equation ∆𝑓 𝐻° = 𝑎 + 𝑏𝐺𝐴 for twelve geometric-arithmetic
Indices1.
No 𝐺𝐴 index 𝑎 𝑏 𝑅2 𝐹 𝑠 𝑄2 𝑆𝐷𝐸𝑃

-385.6264 -30.3508
1h 𝐺𝐴1 0.9970 5989.068 10.0878 0.9962 10.8284
(±4.9250) (±0.3922)

-379.1959 -30.1233
2h 𝐺𝐴4 0.9972 6463.31 9.7118 0.9965 10.2969
(±4.8123) (±0.3747)

-378.328 -30.181
3h 𝐺𝐴6𝑎 0.9973 6581.389 9.6245 0.9966 10.2119
(±4.7790) (±0.3720)

-377.7639 -30.2505
4h 𝐺𝐴6𝑏 0.9974 6818.995 9.4558 0.9967 10.0279
(±4.7006) (±0.3663)

-381.1407 -29.9850
5h 𝐺𝐴6𝑐 0.9966 5269.862 10.752 0.9957 11.5199
(±5.3052) (±0.4131)

-384.6374 -30.3259
6h 𝐺𝐴6𝑑 0.997 6049.15 10.0378 0.9962 10.7668
(±4.9118) (±0.3899)

𝐺𝐴6𝑒 -379.9932 -30.4219

7h 0.9974 7027.406 9.3149 0.9968 9.9111
(β=9.98) (±4.6067) (±0.3629)

𝐺𝐴6𝑓 -376.2199 -30.2804

8h 0.9974 7003.77 9.3305 0.9968 9.8741
(α=0.0025) (±4.6547) (±0.3618)

𝐺𝐴7𝑎 -376.2198 -30.2804

9h 0.9974 7003.78 9.3305 0.9968 9.8741
(β=0.005) (±4.6547) (±0.3618)

𝐺𝐴7𝑏 -376.2198 -30.2804

10h 0.9974 7003.78 9.3305 0.9968 9.8741
(α=0.0025) (±4.6547) (±0.3618)

-379.3122 -30.6458
11h 𝑮𝑨𝟖 0.9975 7199.21 9.2033 0.9969 9.799
(±4.5586) (±0.3612)

-379.7984 -30.4641
12h 𝐺𝐴9 0.9974 6998.026 9.3343 0.9968 9.9293
(±4.6184) (±0.3642)
1
The best model is in bold.
 -46-

The model based on 𝐺𝐴8 index has the best descriptive properties. In the case of this model,
the improvement in the standard deviation is equal to 8.77 % versus the model of Eq 1h. This
model elucidates more than 99.75 % of the variance in the experimental values of ∆𝑓 𝐻 ° of 20
monocarboxylic acids. Table 31 presents the values of 𝐺𝐴8 index, the experimental enthalpies
of formation of 20 monocarboxylic acids as well as the calculated (with Eq 11h) values of
∆𝑓 𝐻 ° for this set of compounds. In the case of the model of Eq 11h, the y-randomization (after
2
1000 repetitions) produced the average value of 𝑅𝑦𝑟𝑎𝑛𝑑 equal to 0.0551 and the average value
2
of 𝑄𝑦𝑟𝑎𝑛𝑑 equal to -0.1734. Thus, it can be claimed that the above model is devoid of any
chance correlations. The results of external validation of the model based on 𝐺𝐴8 descriptor
are listed in Table 29. These values indicate that the above model has a very good ability to
predict external data.

The correlation between the calculated enthalpies of formation of 20 monocarboxylic acids

and the experimental data is presented in Figure 14. In summary, it can be said that the model
of Eq 8h is first-class.

Table 29. Results of external validation of the model based on 𝐺𝐴8 index.
Training set Prediction set 𝑠 𝑅2
BC A 11.8822 0.9985
AC B 14.6203 0.9960
AB C 9.2483 0.9985

Average 11.9169 0.9977

Figure 14. Plot of calculated enthalpies of formation (∆𝑓 𝐻° ) of 20 monocarboxylic acids versus experimental
data.
 -47-

The enthalpies of formation of monocarboxylic acids were modelled using the Randić and
Harary indices by F. Shafiei [46]. But the dataset from [46] does not include formic acid.
Also, this reference does not report the values of 𝑠 for the obtained models. Nevertheless, it
can be established here that for the dataset from Table 31, the single regression for ∆𝑓 𝐻 °
produces the models with 𝑠=14.223 and 𝑅 2 =0.9940 (using H index) and with 𝑠=10.3252 and
𝑅 2 =0.9969 (using χ index). If we discard from this dataset formic acid, the single regression
for ∆𝑓 𝐻 ° gives the models with 𝑠=8.6373 and 𝑅 2 = 0.9967 (using 𝐺𝐴8 index), with 𝑠=9.9256
and 𝑅 2 =0.99682 (using H index) and with 𝑠=9.3424 and 𝑅 2 =0.99713 (using χ index).
Consequently, it can be seen that regardless of whether the dataset contains formic acid or
not, the models based on 𝐺𝐴8 index have lower values of the standard deviation.

4.9 Correlations to the enthalpies of combustion of monocarboxylic acids

Our initial studies have revealed that the enthalpies of combustion of 20 monocarboxylic
acids can be adequately modelled by the single regression. Consequently, we obtained twelve
linear regression equation of the general form ∆𝑐 𝐻 ° = 𝑎 + 𝑏𝐺𝐴. The statistical parameters of
these models are included in Table 30. With respect to the goodness of fit, the models from
Table 30 can be put in the following order:

Eq 3i (𝐺𝐴6𝑎 ) > Eq 10i (𝐺𝐴7𝑏 (α=0.745)) > Eq 4i (𝐺𝐴6𝑏 ) > Eq 8i (𝐺𝐴6𝑓 (α=0.39)) > Eq 9i
(𝐺𝐴7𝑎 (β=0.63)) > Eq 7i (𝐺𝐴6𝑒 (β=0.205)) > Eq 2i (𝐺𝐴4 ) > Eq 12i (𝐺𝐴9 ) > Eq 11i (𝐺𝐴8 ) > Eq
6i (𝐺𝐴6𝑑 ) > Eq 1i (𝐺𝐴1 ) > Eq 5i (𝐺𝐴6𝑐 ).

The best results are obtained by the model based on 𝐺𝐴6𝑎 index. In this case, the
improvement in the standard deviation is equal to 78.98 % compared to the model of Eq 1i.
This model explains more than 99.99 % of the variance in the experimental enthalpies of
combustion of 20 monocarboxylic acids.
Table 30. Regression and statistical parameters of equation ∆𝑐 𝐻° = 𝑎 + 𝑏𝐺𝐴 for twelve geometric-arithmetic
Indices1.
No 𝐺𝐴 index 𝑎 𝑏 𝑅2 𝐹 𝑠 𝑄2 𝑆𝐷𝐸𝑃

855.2810 -650.691
1i 𝐺𝐴1 >0.9999 275039.1 31.9143 >0.9999 36.2354
(±15.581) (±1.241)

992.5140 -645.758
2i 𝐺𝐴4 >0.9999 1098804 15.9673 >0.9999 17.6785
(±7.912) (±0.616)

2
𝑅2 =0.9971 [46].
3
𝑅2 =0.9981 [46].
 -48-

1010.9791 -646.9847
3i 𝑮𝑨𝟔𝒂 >0.9999 6224582 6.7087 >0.9999 6.9075
(±3.3308) (±0.2593)

1022.7190 -648.442
4i 𝐺𝐴6𝑏 >0.9999 4376603 8.007 >0.9999 8.2433
(±3.9770) (±0.310)

952.770 -642.963
5i 𝐺𝐴6𝑐 0.9999 155842 42.3964 0.9998 47.3893
(±20.919) (±1.629)

876.412 -650.150
6i 𝐺𝐴6𝑑 >0.9999 323477.2 29.4281 >0.9999 33.3352
(±14.4000) (±1.143)

𝐺𝐴6𝑒 1040.3737 -649.2262

7i >0.9999 3757957 8.6341 >0.9999 8.9478
(β=0.205) (±4.3003) (±0.3349)

𝐺𝐴6𝑓 1036.9111 -649.1207

8i >0.9999 4040626 8.3267 >0.9999 8.6069
(α=0.39) (±4.1456) (±0.3229)

𝐺𝐴7𝑎 1038.7054 -649.2316

9i >0.9999 3833134 8.5491 >0.9999 8.8403
(β=0.63) (±4.2572) (±0.3316)

𝐺𝐴7𝑏 1020.7715 -648.5517

10i >0.9999 5914927 6.8821 >0.9999 7.0205
(α=0.745) (±3.4205) (±0.2667)

988.849 -656.855
11i 𝐺𝐴8 >0.9999 366752.9 27.6375 >0.9999 29.8392
(±13.689) (±1.085)

978.7565 -652.9905
12i 𝐺𝐴9 >0.9999 569048.8 22.1878 >0.9999 24.1843
(±10.9781) (±0.8656)
1
The best model is in bold.
The values of 𝐺𝐴6𝑎 invariant, the experimental enthalpies of combustion of 20
monocarboxylic acids as well as the calculated (with Eq 3i) values of ∆𝑐 𝐻 ° for this set of
compounds are listed in Table 31. In the case of the model of Eq 3i, the y-scrambling (after
2 2
1000 repetitions) gave the average values of 𝑅𝑦𝑟𝑎𝑛𝑑 and 𝑄𝑦𝑟𝑎𝑛𝑑 equal to 0.0536 and -0.1768,
respectively. The results of external validation of the model based on 𝐺𝐴6𝑎 index are
contained in Table 32.
Table 31. Experimental and calculated (with Eq 11h or Eq 3i) the enthalpies of formation (∆𝑓 𝐻° ) and the
enthalpies of combustion (∆𝑐 𝐻° ) of 20 monocarboxylic acids with values of 𝐺𝐴8 and 𝐺𝐴6𝑎 indices.
∆𝑓 𝐻° (kJ/mol) ∆𝑐 𝐻 ° (kJ/mol)
Subset Compound 𝐺𝐴10 𝐺𝐴6
Exptl Calcd Exptl Calcd
B formic acid -425.5 -439.98 1.979 -253.8 -256.85 1.960
A ethanoic acid -484.5 -467.42 2.875 -874.2 -868.34 2.905
C propanoic acid -510.8 -498.48 3.889 -1527.3 -1526.28 3.922
A butanoic acid -533.9 -527.75 4.844 -2183.5 -2184.75 4.939
B pentanoic acid -558.9 -557.37 5.810 -2837.8 -2839.69 5.952
B hexanoic acid -583.58 -587.26 6.786 -3492.4 -3492.07 6.960
A heptanoic acid -608.5 -617.35 7.767 -4146.9 -4142.86 7.966
C octanoic acid -634.8 -647.56 8.753 -4799.9 -4792.63 8.970
 -49-

B nonanoic acid -658 -677.86 9.742 -5456.1 -5441.74 9.974

B decanoic acid -714.1 -708.23 10.733 -6079.3 -6090.40 10.976
C undecanoic acid -736.2 -738.65 11.726 -6736.5 -6738.74 11.978
B dodecanoic acid -775.1 -769.11 12.719 -7377 -7386.84 12.980
A tridecanoic acid -807.2 -799.60 13.714 -8024.2 -8034.76 13.981
B tetradecanoic acid -834.1 -830.11 14.710 -8676.7 -8682.55 14.983
A pentadecanoic acid -862.4 -860.63 15.706 -9327.7 -9330.24 15.984
A hexadecanoic acid -892.2 -891.18 16.703 -9977.2 -9977.83 16.985
C heptadecanoic acid -924.4 -921.73 17.700 -10624.4 -10625.35 17.985
A octadecanoic acid -948 -952.30 18.697 -11280.1 -11272.82 18.986
C nonadecanoic acid -984.1 -982.87 19.695 -11923.4 -11920.24 19.987
C eicosanoic acid -1012.6 -1013.45 20.693 -12574.2 -12567.61 20.987

The high average value of 𝑅 2 and the low average value of 𝑠 testify that the above model is
reliable with respect to external data. From the plot in Figure 15, it can be inferred that the
calculated values (with Eq 3i) are in agreement with the experimental enthalpies of
combustion of 20 monocarboxilic acids.

Table 32. Results of external validation of the model based on 𝐺𝐴6𝑎 index.
Training set Prediction set 𝑠 𝑅2
BC A 6.8620 >0.9999
AC B 10.3101 >0.9999
AB C 7.0395 >0.9999

Average 8.0705 >0.9999

Figure 15 and all statistical metrics suggest that the model of Eq 3i is very good. The
enthalpies of combustion of monocarboxylic acids were modelled using the Randić and
Harary indices by F. Shafiei [46]. But in [46], formic acid is excluded from studies. Also, this
reference does not contain the values of the standard deviation for the models based on these
invariants.

Figure 15. Plot of calculated enthalpies of combustion (∆𝑐 𝐻° ) of 20 monocarboxylic acids versus experimental
data.
 -50-

Nevertheless, it can be demonstrated here, that for the dataset from Table 31, the single
regression for ∆𝑐 𝐻 ° produces the model with 𝑠=248.075 and 𝑅 2 =0.9946 (using H index) and
with 𝑠=40.5239 and 𝑅 2 =0.9999 (using χ index). If we remove from this dataset formic acid,
the single regression for ∆𝑐 𝐻 ° produces the models with 𝑠=6.8545 and 𝑅 2 >0.9999 (using
𝐺𝐴6𝑎 index), with 𝑠=248.075 and 𝑅 2 =0.99564 (using H index) and with 𝑠=10.5151 and
𝑅 2 >0.99995 (using χ index). Therefore, it can be seen that regardless of if the dataset includes
formic acid or not, the models based on 𝐺𝐴6𝑎 index have lower values of the standard
deviation.

In the above paragraphs (b-i), we have presented eight QSPR models based on the newly
defined topological indexes. All final models exhibited the values of 𝑅 2 and 𝑄 2 above 0.99
and relatively low values of 𝑠. In all cases, the best statistical parameters were possessed by
the model based on any of the newly defined molecular descriptors. Namely, the
improvement in the standard deviation was in the range of 8.77 % (Eq 11h) to 85.82 % (Eq
10c) compared to the models based on the first geometric-arithmetic index. In four cases, the
best models were based on the sixth geometric-arithmetic index (on its 𝐺𝐴6𝑑 (Eq 6b), 𝐺𝐴6𝑓
(at α=0.235, Eq 8e), 𝐺𝐴6𝑓 (at α=0.315, Eq 8f) and 𝐺𝐴6𝑎 (Eq 3i) versions). In three cases, the
best models were based on the seventh geometric-arithmetic index (on its 𝐺𝐴7𝑏 (at α=0.605,
Eq 10c), 𝐺𝐴7𝑎 (at β=0.005 Eq 9d) or 𝐺𝐴7𝑏 (at α=0.0025, Eq 10c) and 𝐺𝐴7𝑏 (at α=0.97 Eq
10g) versions). In one case, the best model was based on the eighth geometric-arithmetic
index (Eq 11h). On the other hand, in four cases the worst models were based on the first
geometric-arithmetic index (Eq 1c, Eq 1d, Eq 1e and Eq 1f). In two cases, the worst models
were based on the fourth geometric-arithmetic index (Eq 2a and Eq 2g). Also in two cases, the
worst models were based on the sixth geometric-arithmetic index (on its 𝐺𝐴6𝑐 (Eq 5h and Eq
5i) version). All final models explain more than 99 % of the variance in the experimental data.
In all cases, the y-randomization (after 1000 repetitions) applied to the final models gave the
average value of 𝑅 2 from 0.0229 (Eq 8f) to 0.0562 (Eq 10g) and the average value of 𝑄 2
from -0.2065 (Eq 10g) to -0.0703 (Eq 8f). Therefore, it can be claimed that all final models
are devoid of any chance correlations. In the procedure of external validation, all predictions
were made with the value of 𝑅 2 above 0.99. Hence, all final models have very good
predictive capabilities relative to external data.

4
𝑅2 =0.9934 [46].
5
𝑅2 =1 [46].
 -51-

In conclusion, it can be stated that all the above presented models according to the criteria
cited in Part Two are excellent and reliable with respect to external data.

5 Concluding remarks

In this work, we have introduced several new geometric-arithmetic indices. We have

demonstrated that these newly defined descriptors have an extremely low level of degeneracy
(Part Three) as well as in many cases exhibit better correlation properties than the first
geometric-arithmetic index (Part Four).

It is hoped that the newly proposed molecular invariants will be widely used in QSAR/QSPR
studies.

References

[1] N. Adriaanse, H. Dekker, J. Coops, Heats of combustion of normal saturated fatty

acids and their methyl esters, Recl. Trav. Chim. Pays-Bas 84 (1965) 393-407.
[2] A. T. Balaban, O. Ivanciuc, Historical development of topological indices, in: J.
Devillers, A. T. Balaban (Eds.), Topological Indices and Related descriptors in QSAR
and QSPR, Gordon & Breach, Amsterdam, 1999, pp. 21-57.
[3] M. Benzi, C. Klymko, Total communicability as a centrality measure, J. Complex
Networks 1 (2013) 124-149.
[4] J. Chao, F. D. Rossini, Heats of combustion, formation and isomerization of nineteen
alkanols, J. Chem. Eng. Data 10 (1965) 374-379.
[5] M. J. Crawley, Statistics. An Introduction Using R, Wiley, Chichester, 2015.
[6] J. J. Crofts, D. H. Higham, A weighted communicability measure applied to complex
brain networks, J. R. Soc. Interface 6 (2009) 411-414.
[7] G. Csardi, T. Nepusz, The igraph software package for complex network research,
InterJournal Complex Systems 1695 (2006) http://igraph.org
[8] K. C. Das, On geometric-arithmetic index of graph, MATCH Commun. Math. Comput.
Chem. 64 (2010) 619-630.
[9] K. C. Das, I. Gutman, B. Furtula, On second geometric-arithmetic index of graphs,
Iran. J. Math. Chem. 1 (2010) 17-27.
[10] K. C. Das, I. Gutman, B. Furtula, On third geometric-arithmetic index of graphs, Iran.
J. Math. Chem. 1 (2010) 29-36.
 -52-

[11] K. C. Das, I. Gutman, B. Furtula, Survey on geometric-arithmetic indices of graphs,

MATCH Commun. Math. Comput. Chem. 65 (2011) 595-644.
[12] K. C. Das, I. Gutman, B. Furtula, On first geometric-arithmetic index of graphs, Discr.
Appl. Math. 159 (2011) 2030-2037.
[13] K. C. Das, N. Trinajstić, comparison between geometric-arithmetic indices, Croat.
Chim. Acta 85 (2012) 353-357.
[14] M. Dehmer, M. Grabner, B. Furtula, Structural discrimination of networks by using
distance, degree and eigenvalue-based measures, PLoS ONE 7 (2012) 1-15.
[15] A. A. Dobrynin, A. A. Kocheova, Degree distance of a graph: a degree analog of the
Wiener index, J. Chem. Inf. Comput. Sci. 34 (1994) 1082-1086.
[16] E. Estrada, J. A. Rodriguez-Velazquez, Subgraph centrality in complex networks,
Phys. Rev. E 71 (2005) #056103.
[17] E. Estrada, Virtual identification of essential proteins within the protein interaction
network of yeast, Proteomics 6 (2006) 35-40.
[18] E. Estrada, Protein bipartivity and essentiality in the yeast protein-protein interaction
network, J. Proteome Res. 5 (2006) 2177-2184.
[19] G. Fath-Tabar, B. Futula, I. Gutman, A new geometric-arithmetic index, J. Math.
Chem. 47 (2010) 477-486.
[20] B. Freeman, M. O. Bagby, Heats of combustion of fatty esters and triglycerides, J.
Am. Oil Chem. Soc. 66 (1989) 1601-1605.
[21] M. Ghorbani, A. Khaki, A note on the fourth version of geometric-arithmetic index,
Optoelectron. Adv. Mater. Rapid Commun. 4 (2010) 2212-2215.
[22] A. Graovac, M. Ghorbani, M. A. Hosseinzadeh, Computing fifth geometric-arithmetic
index for nanostar dendrimers, J. Math. Nanosci. 1 (2011) 33-42.
[23] R. Ihaka, R. Gentleman, R: a language for data analysis and graphics, J. Comput.
Graph. Stat. 5 (1996) 299-314.
[24] O. Ivanciuc, Graph theory in chemistry, in: J. Gasteiger (Ed.), Handbook of
Chemoinformatics, Wiley-VCH, Weinheim, 2003, pp. 103-138.
[25] O. Ivanciuc, Topological indices, in: J. Gasteiger (Ed.), Handbook of
Chemoinformatics, Wiley-VCH, Weinheim, 2003, pp. 981-1003.
[26] O. Ivanciuc, A. T. Balaban, The graph description of chemical structures, in: J.
Devillers, A. T. Balaban (Eds.), Topological Indices and Related descriptors in QSAR
and QSPR, Gordon & Breach, Amsterdam, 1999, pp. 59-167.
 -53-

[27] D. Janežič, A. Miličević, S. Nikolić, N. Trinajstić, Graph Theoretical Matrices in

Chemistry, Univ. Kragujevac, Kragujevac, 2007.
[28] B. S. Junkes, R. D. M. C. Amboni, R. A. Yunes, V. E. F. Heinzen, Prediction of the
chromatographic retention of saturated alcohols on stationary phases of different
polarity applying the novel semi-empirical topological index, Anal. Chim. Acta 477
(2003) 29-39.
[29] R. Kiralj, M. M. C. Ferreira, Basic validation procedures for regression models in
QSAR and QSPR studies: theory and application, J. Braz. Chem. Soc. 20 (2009) 770-
787.
[30] C. F. Klymko, Centrality and Communicability Measures in Complex Networks:
Analysis and Algorithms, PhD thesis, Emory University, Atlanta, GA 2013.
[31] E. V. Konstantinova, V. A. Skorobogatov, M. V. Vidyuk, Applications of information
theory in chemical graph theory, Indian J. Chem. 42A (2003) 1227-1240.
[32] N. D. Lebedeva, Heats of combustion of monocarboxylic acids, Russ. J. Phys. Chem.
(Engl. Transl.) 38 (1964) 1435-1437.
[33] A. Mahmiani, O. Khormali, A. Iranmanesh, On the edge version of geometric-
arithmetic index, Dig. J. Nanomater. Biostruct. 7 (2012) 411-414.
[34] A. Mahmiani, O. Khormali, On the total version of geometric-arithmetic index, Iran.
J. Math. Chem. 4 (2013) 21-26.
[35] Z. Mihalić, N. Trinajstić, A graph-theoretical approach to structure-property
relationships, J. Chem. Educ. 69 (1992) 701-712.
[36] C. Mosselman, H. Dekker, Enthalpies of formation of n-alkan-1-ols, J. Chem. Soc.,
Faraday Trans. 71 (1975) 417-424.
[37] L. Mu, C. Feng, L. Xu, Study on QSPR of alcohols with a novel edge connectivity
index 𝑚𝐹 , MATCH Commun. Math. Comput. Chem. 56 (2006) 217-230.
[38] NIST Chemistry WebBook, http://webbook.nist.gov/chemistry/
[39] A. Platzer, P. Perco, A. Lukas, B. Mayer, Characterization of protein-interaction
networks in tumors, BMC Bioinf. 8 (2007) 224.
[40] G. Pólya, R. C. Read, Combinatorial Enumeration of Groups, Graphs and Chemical
Compounds, Springer, New York, 1987.
[41] R Core Team (2015), R: a language and environment for statistical computing. R
Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-
project.org/
 -54-

[42] B. Ren, A new topological index for QSPR of alkanes, J. Chem. Inf. Comput. Sci. 39
(1999) 139-143.
[43] B. Ren, New atom-type-based AI topological indices: application to QSPR studies of
aldehydes and ketones, J. Comput. Aided Mol. Des. 17 (2003) 607-620.
[44] J. M. Rodríguez, J. M. Sigarreta, On the geometric-arithmetic index, MATCH
Commun. Math. Comput. Chem. 74 (2015) 103-120.
[45] J. M. Rodríguez, J. M. Sigarreta, Spectral study of the geometric-arithmetic index,
MATCH Commun. Math. Comput. Chem. 74 (2015) 121-135.
[46] F. Shafiei, Relationship between topological indices and thermodynamic properties
and of the monocarboxylic acids applications in QSPR, Iran. J. Math. Chem. 6 (2015)
15-28.
[47] H. A. Skinner, A. Snelson, The heats of combustion of the four isomeric butyl
alcohols, Trans. Faraday Soc. 56 (1960) 1776-1783.
[48] R. Todeschini, V. Consonni, Molecular Descriptors for Chemoinformatics, Wiley-
VCH, Weinheim, 2009.
[49] D. Vukičević, B. Furtula, Topological index based on the ratios of geometrical and
arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46 (2009) 1369-
1376.
[50] B. Zhou, I. Gutman, B. Furtula, Z. Du, On two types of geometric-arithmetic index,
Chem. Phys. Lett. 482 (2009) 153-155.