May 11, 2018 14:26 WSPC/S1793-8309 257-DMAA 1850037

Discrete Mathematics, Algorithms and Applications

Vol. 10, No. 3 (2018) 1850037 (16 pages)


c World Scientific Publishing Company
DOI: 10.1142/S1793830918500374

Entire Zagreb indices of graphs

Anwar Alwardi
DOS in Mathematics, University of Mysore
Mysore 570 006, India
Department of Mathematics, College of Education
Yafea, University of Aden, Aden, Yemen
a wardi@hotmail.com

Akram Alqesmah∗ and R. Rangarajan†

DOS in Mathematics, University of Mysore
Mysore 570 006, India
∗aalqesmah@gmail.com
†rajra63@gmail.com

Ismail Naci Cangul

Uludag University, Mathematics
Gorukle 16059, Bursa, Turkey
ncangul@gmail.com

Received 24 June 2017

Accepted 3 March 2018
Published 2 April 2018

The Zagreb indices have been introduced in 1972 to explain some properties of chemical
compounds at molecular level mathematically. Since then, the Zagreb indices have been
studied extensively due to their ease of calculation and their numerous applications in
place of the existing chemical methods which needed more time and increased the costs.
Many new kinds of Zagreb indices are recently introduced for several similar reasons.
In this paper, we introduce the entire Zagreb indices by adding incidency of edges and
vertices to the adjacency of the vertices. Our motivation in doing so was the following
fact about molecular graphs: The intermolecular forces do not only exist between the
atoms, but also between the atoms and bonds, so one should also take into account the
relations (forces) between edges and vertices in addition to the relations between vertices
to obtain better approximations to intermolecular forces. Exact values of these indices
for some families of graphs are obtained and some important properties of the entire
Zagreb indices are established.

Keywords: First entire Zagreb index; second entire Zagreb index.

Mathematics Subject Classification 2010: 05C07, 05C30, 05C76

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1. Introduction
A graph is a collection of points together with a number of lines connecting a subset
of them. The points and lines of a graph are called vertices and edges of the graph,
respectively. The vertex and edge sets of a graph G are denoted by V (G) and
E(G), or briefly by V and E, respectively. If we think of molecules as particular
chemical structures, and if we replace atoms and bonds with vertices and edges,
respectively, the obtained graph is called a molecular graph. That is, a molecular
graph is a simple graph such that its vertices correspond to the atoms and its
edges to the bonds. Note that hydrogen atoms are often omitted and the remaining
part of the graph is sometimes called as the carbon graph of the corresponding
molecule. Chemical graph theory which deals with the above mentioned relations
between molecules and corresponding graphs is a branch of mathematical chemistry
which has an important effect on the development of the molecular chemistry and
QSAR/QSPR studies.
In this paper, we are concerned with simple graphs which are finite, undirected
with no loops nor multiple edges. Throughout this paper, we let |V (G)| = p and
|E(G)| = q. The complement of G, denoted by G, is a simple graph on the same set
of vertices V (G) in which two vertices u and v are adjacent if and only if they are not
adjacent
inG. That is, uv ∈ E(G) ⇔ uv ∈ / E(G). Obviously, E(G)∪E(G) = E(Kp )
and q = p2 − q, where q denotes the number of edges of G.
The open and closed neighborhoods of a vertex v are denoted by N (v) = {u ∈
V : uv ∈ E} and N [v] = N (v) ∪ {v}, respectively. The degree of a vertex v in G is
denoted by degG (v) or briefly by dG (v), and is defined to be the number of edges
incident with v. In fact, deg(v) = |N (v)|. When there is no confusion, one can also
omit G and use d(v) instead of dG (v). Clearly, the degree of the same vertex v in
G is given by degG (v) = p − 1 − deg(v). The minimum degree of G is denoted by
δ, and the maximum degree is denoted by ∆. If δ = ∆ = k for a graph G, we say
that G is a regular graph of degree k. Also a graph with the property that ∆ ≤ 4
is called a molecular graph.
The line graph of a simple graph G is a graph denoted by L(G) that represents
the adjacencies between the edges of G. In other words, L(G) is the graph of which
the vertices are the edges of G. The total graph T (G) of a graph G is the graph
such that the vertex set of T (G) consists of the vertices and edges of G and two
vertices are adjacent in T (G) if and only if their corresponding elements are either
adjacent or incident in G. It is easy to see that |V (T (G))| = p + q and |E(T (G))| =

2q + 12 v∈V (G) [d(v)]2 .
As usual, the path, wheel, cycle, star, complete graphs with p vertices are
denoted by Pp , Wp , Cp , Sp , Kp ; and the complete bipartite and tadpole graphs
are denoted by Kr,s and Tr,s , respectively. For more detailed information about the
definitions and terminologies about graphs, see [7].
A topological graph index is a numerical value associated with chemical
constitution purporting for correlation of chemical structure with various physical

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properties, chemical reactivity or biological activity. In an exact phrase, if Gr

denotes the class of all finite graphs then a topological index is a function Top
from Gr into real numbers with this property that Top(G) = Top(H), if G and
H are isomorphic. Obviously, the number of vertices and the number of edges
are two basic topological indices. In recent decades, a large number of topolog-
ical indices have been defined and utilized for chemical documentation, isomer
discrimination, study of molecular complexity, chirality, similarity/dissimilarity,
QSAR/QSPR, drug design and database selection, lead optimization, etc.
As an example, the boiling point of a molecule is directly related to the forces
between the atoms. When a solution is heated, the temperature is increased and as
it is increased, the kinetic energy between molecules increases. This means that the
molecular motion becomes so intense that the bonds between molecules break and
become a gas. The moment the liquid turns to gas is labeled as the boiling point.
The boiling point can give important clues about the physical properties of chemical
structures. Molecules which strongly interact or bond with each other through a
variety of intermolecular forces cannot move easily or rapidly and therefore, do not
achieve the kinetic energy necessary to escape the liquid state. That is why the
boiling points of the alkanes increase with molecular size.
Two of the most useful topological graph indices are the first and second Zagreb
indices which have been introduced by Gutman and Trinajstic in [6]. They are
denoted by M1 (G) and M2 (G) and were defined as

M1 (G) = [d(u)]2
u∈V (G)

and

M2 (G) = d(u)d(v),
uv∈E(G)

respectively. In [10], it was shown that

  
M1 (G) = d(v) = [d(u) + d(v)],
u∈V (G) v∈N (u) uv∈E(G)

and
1  
M2 (G) = d(u) d(v).
2
u∈V (G) v∈N (u)

Similarly, the first and second Zagreb coindices of a graph G are denoted by
M1 (G) and M2 (G), respectively, and were introduced in [3], as

M1 (G) = [degG (u) + degG (v)]
uv∈E(G)

and

M2 (G) = degG (u)degG (v).
uv∈E(G)

Some properties of the Zagreb coindices of some graph operations are studied in [1].

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The Zagreb indices have been studied extensively. Many new reformulated and
extended versions of the Zagreb indices have been introduced, e.g. see [2, 4, 5, 8–14].

2. Some Properties of the Entire Zagreb Indices

Now we define the entire Zagreb indices and then obtain some fundamental prop-
erties of them. The chemical reason for defining these new indices is that the inter-
molecular forces do not only exist between the atoms of a molecule but also between
the atoms and bonds, so one should also take into account the relations (forces)
between the edges and vertices in addition to the relations between vertices. We
obtain exact values of these newly defined indices for some families of graphs and
also for some graph operations, and establish some important properties of the
entire Zagreb indices.
Let G = (V, E) be a graph, where V and E are the vertex and edge sets,
respectively. In this paper, by a word element in a graph G, we mean either a vertex
or an edge. Now, we define the first and second entire Zagreb indices as follows:
Definition 2.1. Let G = (V, E) be a graph. Then the first and second entire Zagreb
indices are defined by

M1E (G) = (deg(x))2 ,
x∈V (G)∪E(G)

and

M2E (G) = deg(x)deg(y).
x is either adjacent
or incident to y

Example 2.2. Let G be a graph as in Fig. 1,

(1)
 
4 
4
M1E (G) = (deg(x)) = 2 2
(deg(vi )) + (deg(ei ))2
x∈V (G)∪E(G) i=1 i=1

2 2 2 2 2 2 2 2
= 1 + 3 + 2 + 2 + 2 + 3 + 2 + 3 = 44.

v1
u
e1
v2 u
e2 @ e4
@
e3 @
v3 v @u v4

Fig. 1. Graph G.

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Entire Zagreb indices of graphs

(2)

M2E (G) = deg(x)deg(y)
x is either adjacent
or incident to y

= deg(v1 )deg(v2 ) + deg(v2 )deg(v3 ) + deg(v3 )deg(v4 ) + deg(v4 )deg(v2 )

+ deg(e1 )deg(e2 ) + deg(e1 )deg(e4 ) + deg(e2 )deg(e3 )
+ deg(e2 )deg(e4 ) + deg(e3 )deg(e4 ) + deg(v1 )deg(e1 )
+ deg(v2 )deg(e1 ) + deg(v2 )deg(e2 ) + deg(v2 )deg(e4 )
+ deg(v3 )deg(e2 ) + deg(v3 )deg(e3 ) + deg(v4 )deg(e3 )
+ deg(v4 )deg(e4 ) = 98.

According to Definition 2.1 and the definition of Zagreb indices, the first and
second entire Zagreb indices can be expressed in terms of the first and second Zagreb
indices as follows:

Theorem 2.3. For any graph G, the first and second entire Zagreb indices are
equal to

M1E (G) = 4q − 3M1 (G) + 2M2 (G) + (deg(u))3
u∈V (G)

and
1 
M2E (G) = 4q − 2M1E (G) − 2M1 (G) + M2 (G) + (deg(u))4
2
u∈V (G)
 
+ (deg(u))2 deg(v)
u∈V (G) v∈N (u)
 2
1  
+  deg(v) .
2
u∈V (G) v∈N (u)

Proof. Let G be a graph with |V (G)| = p and |E(G)| = q. Then

M1E (G) = M1 (G) + M1 (L(G))

= [deg(u) + deg(v) + (deg(uv))2 ]
uv∈E(G)

= 4q − 3M1 (G) + 2M2 (G) + [(deg(u))2 + (deg(v))2 ]
uv∈E(G)

= 4q − 3M1 (G) + 2M2 (G) + (deg(u))3 .
u∈V (G)

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Similarly,

M2E (G) = deg(x)deg(y)
x is either adjacent
or incident to y
 
= deg(u)deg(v) + deg(e1 )deg(e2 )
uv∈E(G) e1 e2 ∈E(L(G))

+ deg(v)deg(e)
v is incident to e
 
= M2 (G) + M2 (L(G)) + deg(u) deg(uv).
u∈V (G) v∈N (u)

Since,
 
1     
M2 (L(G)) = deg(uv) 
 deg(uw) + deg(vz)

2
uv∈E(G) w∈N (u) z∈N (v)
w=v z=u

1 
= 5M1 (G) − 3M2 (G) − 4q + [(deg(u))3 + (deg(v))3 ]
2
uv∈E(G)

1 
+ deg(u)deg(v)[deg(u) + deg(v)]
2
uv∈E(G)
 
1     
+ deg(uv) 
 deg(w) + deg(z)

2
uv∈E(G) w∈N (u) z∈N (v)
w=v z=u

5 
− [(deg(u))2 + (deg(v))2 ]
2
uv∈E(G)

1 
= 5M1 (G) − 6M2 (G) − 4q + (deg(u))3 (deg(u) − 5)
2
u∈V (G)
 
+ [(deg(u))2 + 1] deg(v)
u∈V (G) v∈N (u)
 2 
1     
+  deg(v) − (deg(v))2 
2
u∈V (G) v∈N (u) v∈N (u)

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1 
= 8q − 3M1 (G) − 3M1E (G) + (deg(u))4
2
u∈V (G)

 2
  1  
+ (deg(u)) 2
deg(v) +  deg(v) ,
2
u∈V (G) v∈N (u) u∈V (G) v∈N (u)
  2
 3
where u∈V (G) v∈N (u) (deg(v)) = u∈V (G) (deg(u))
, and since,
  
deg(u) deg(uv) = 2M2 (G) − 2M1 (G) + [deg(u)]3
u∈V (G) v∈N (u) u∈V (G)

= M1E (G) + M1 (G) − 4q,

the result is obtained.

Now it is the turn to give some relations between the entire Zagreb indices and
entire Zagreb coindices:

Proposition 2.4. Let G be a graph. Then

M1E (G) = (5p − 12)M1 (G) − M1E (G) + 2(p(p − 1) + 2q 2 ) + 2(p − 1)2 (q − 5q)
+ p(p − 1)3 − 6(p − 1)(q − q).

Proof. By Theorem 2.3, we have


M1E (G) = 4q − 3M1 (G) + 2M2 (G) + (degG (u))3 .
u∈V (G)

Since,
M1 (G) = M1 (G) + 2(p − 1)(q − q),
and
1
M2 (G) = (2p − 3)M1 (G) − M2 (G) + (p − 1)2 (q − 2q) + 2q 2
2
by [1], and
 
(degG (u))3 = (p − 1 − deg(u))3 = p(p − 1)3 − 6q(p − 1)2
u∈V (G) u∈V (G)

+ 3(p − 1)M1 (G) − (deg(u))3
u∈V (G)

= 3(p − 2)M1 (G) − M1E (G) + 2M2 (G) + p(p − 1)3

− 6q(p − 1)2 + 4q,

the result follows.

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Note that, Proposition 2.4 can also be restated as

M1E (G) + M1E (G) = (5p − 12)M1 (G) + 2(p(p − 1) + 2q 2 ) + 2(p − 1)2 (q − 5q)
+ p(p − 1)3 − 6(p − 1)(q − q).
Proposition 2.5. Let G be a graph. Then
1
M2E (G) = M2E (G) − (3p − 8)M1E (G) + [17p2 − 76p + 93 − 8q]M1 (G) − 2M2 (G)
2
+ (p − 1)3 [2p(p − 2) − 5q] − (p − 1)2 (11pq − 27q + 3q − p) + 4q(3p − 7)
+ 2q(p − 2)(p + 2q − 1) − 2[2q − q 2 (p − 3)] + 2(p − 1)(3q + 4q 2 − 4q).

Proof. From Theorem 2.3, we have

1 
M2E (G) = 4q − 2M1E (G) − 2M1 (G) + M2 (G) + (degG (u))4
2
u∈V (G)
 2
  1  
+ (degG (u))2 degG (v) +  degG (v) .
2
u∈V (G) v∈NG (u) u∈V (G) v∈NG (u)

So, by using
M1 (G) = M1 (G) + 2(p − 1)(q − q),
and
1
M2 (G) = (2p − 3)M1 (G) − M2 (G) + (p − 1)2 (q − 2q) + 2q 2 ,
2
from [1] together with Proposition 2.4 and applying the equalities
degG (w) = p − 1 − deg(w),
 
deg(v) = 2q − deg(u) − deg(v),
v∈NG (u) v∈N (u)

and
 
(deg(v))2 = M1 (G) − (deg(u))2 − (deg(v))2 ,
v∈NG (u) v∈N (u)

the result is obtained.

3. Entire Zagreb Indices for Some Graph Families

In this section, we compute the first and second entire Zagreb indices for some
families of graphs and some graph operations. First, we have a useful result:
Proposition 3.1. Let G be a k-regular graph on p vertices. Then
(1) M1E (G) = pk(2k 2 − 3k + 2).
(2) M2E (G) = pk(2k 3 − 72 k 2 + 4k − 2).

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kp
Proof. Clearly q = 2and also deg(uv) = 2k − 2 for each uv ∈ E(G). Thus,

M1E (G) = [deg(u) + deg(v) + (deg(uv))2 ]
uv∈E(G)

= q(2k + (2k − 2)2 )

= pk(2k 2 − 3k + 2).
Since E(L(G)) = 12 M1 (G) − q = p2 k(k − 1), then
 
M2E (G) = M2 (G) + M2 (L(G)) + deg(u) deg(uv)
u∈V (G) v∈N (u)
 
7 2
= pk 2k − k + 4k − 2 .
3
2

The following are direct results of this proposition:

Corollary 3.2. For any complete graph Kp ,

(1) M1E (Kp ) = p(p − 1)(2p2 − 7p + 7).

p(p−1)
(2) M2E (Kp ) = 2 (4(p − 1)3 − 7(p − 1)2 + 8(p − 1) − 4).

Corollary 3.3. For any cycle graph Cp ,

(1) M1E (Cp ) = 8p

(2) M2E (Cp ) = 16p.

Proposition 3.4. For any path graph Pp with p ≥ 3,

(1) M1E (Pp ) = 8(p − 2).


11, if p = 3;
E
(2) M2 (Pp ) =
2(8p − 19), otherwise.

Proof. Let V (Pp ) = {v1 , v2 , . . . , vp } and E(Pp ) = {e1 , e2 , . . . , ep−1 }. Then the
degrees of all elements of Pp are equal to 2 except the elements v1 , e1 , vp , ep−1
which have degrees equal to one. Thus, by using the formula for the first Zagreb
index of Pp , we have
M1E (Pp ) = M1 (Pp ) + M1 (L(Pp )) = 4p − 6 + 4(p − 1) − 6 = 8(p − 2).
Now, for the second entire Zagreb index, if p = 3, it is easy to see that M2E (P3 ) = 11.
Suppose now, p ≥ 4. Then
M2E (Pp ) = 4(p − 2) + 4(p − 3) + 2 + 4 + 2(p − 3) · 4 = 2(8p − 19),
as deg(v1 )deg(e1 ) = deg(vp )deg(ep−1 ) = 1 and deg(v2 )deg(e1 ) = deg(vp−1 )deg
(ep−1 ) = 2.

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Proposition 3.5. For the complete bipartite graph Kr,m ,

(1) M1E (Kr,m ) = rm[(r + m)2 − 3(r + m) + 4].

(2) M2E (Kr,m ) = 12 rm[(r + m)3 − (r + m)(3(r + m) − 4) − (r − 2)2 − (m − 2)2 ].

Proof. By using Theorem 2.3, we have

(1)

M1E (Kr,m ) = 4rm − 3rm(r + m) + 2r2 m2 + rm3 + mr3

= rm[(r + m)2 − 3(r + m) + 4].

(2)

M2E (Kr,m ) = 4rm − 2rm[(r + m)2 − 3(r + m) + 4] − 2rm(r + m) + r2 m2

1 1
+ rm(r3 + m3 ) + r3 m2 + r2 m3 + rm(r2 m + rm2 )
2 2
1
= rm[(r + m)3 − 4r2 − 4m2 − 6rm + 8r + 8m − 8]
2
1
= rm[(r + m)3 − (r + m)(3(r + m) − 4) − (r − 2)2 − (m − 2)2 ].
2

Corollary 3.6. For the star graph Sp , we have

(1) M1E (Sp ) = (p − 1)[p2 − 3p + 4],

(2) M2E (Sp ) = 12 (p − 1)[p3 − p(3p − 4) − (p − 3)2 − 1].

Proof. Apply Proposition 3.5 for Sp which is isomorphic to K1,p−1 .

4. Entire Zagreb Indices of Join and Corona Products of Two

Graphs
Graph theoretical operations are used to obtain combinatorial properties of large
graphs in terms of smaller graphs. In this section, we deal with two of the most
well-known operations, sum (join) and corona products and calculate the first and
second entire Zagreb indices of the sum and corona products of some graph types.
These results can be extended to other graph operations.

Sum (Join)
The join G1 +G2 of two graphs G1 and G2 with disjoint vertex sets |V (G1 )| = p1 ,
|V (G2 )| = p2 and edge sets |E(G1 )| = q1 , |E(G2 )| = q2 is the graph on the vertex set
V (G1 )∪V (G2 ) and the edge set E(G1 )∪E(G2 )∪{u1 u2 : u1 ∈ V (G1 ); u2 ∈ V (G2 )}.
Hence, the join of two graphs is obtained by connecting each vertex of one graph to

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each vertex of the other graph, while keeping all edges of both graphs. The degree
of any vertex v ∈ G1 + G2 is given by

degG1 (v) + p2 , if v ∈ V (G1 );
degG1 +G2 (v) =
degG2 (v) + p1 , if v ∈ V (G2 ).
First, we recall a result on the first and second Zagreb indices of the sum of two
graphs:
Proposition 4.1 ([8]). For any two graphs G1 and G2 with |V (G1 )| = p1 ,
|V (G2 )| = p2 and |E(G1 )| = q1 , |E(G2 )| = q2 , the first and second Zagreb indices
of G1 + G2 are given by
(1) M1 (G1 + G2 ) = M1 (G1 ) + M1 (G2 ) + 4(p1 q2 + p2 q1 ) + p1 p2 (p1 + p2 ).
(2) M2 (G1 + G2 ) = M2 (G1 ) + M2 (G2 ) + p2 M1 (G1 ) + p1 M1 (G2 ) + p21 q2 + p22 q1 +
4q1 q2 + 2p1 p2 (q1 + q2 ) + p21 p22 .
Now we can state our result on the first and second entire Zagreb indices of the
sum of two graphs:

Theorem 4.2. For any two graphs G1 and G2 with |V (G1 )| = p1 , |V (G2 )| = p2
and |E(G1 )| = q1 , |E(G2 )| = q2 , the first entire Zagreb index of G1 + G2 is given
by
M1E (G1 + G2 ) = M1E (G1 ) + M1E (G2 ) + 5[p2 M1 (G1 ) + p1 M1 (G2 )]
+ p1 p2 [(p1 + p2 )2 − 3(p1 + p2 ) + 4(q1 + q2 ) + 4]
+ 8(p21 q2 + p22 q1 + q1 q2 ) − 12(p1 q2 + p2 q1 ).

Proof. Let G1 and G2 be two graphs with |V (G1 )| = p1 , |V (G2 )| = p2 and

|E(G1 )| = q1 , |E(G2 )| = q2 . Then by Theorem 2.3,
M1E (G1 + G2 ) = 4qG1 +G2 − 3M1 (G1 + G2 ) + 2M2 (G1 + G2 )

+ (degG1 +G2 (u))3 .
u∈V (G1 +G2 )

Since qG1 +G2 = q1 + q2 + p1 p2 and M1 (G1 + G2 ), M2 (G1 + G2 ) are obtained in


Proposition 4.1, it remains to calculate u∈V (G1 +G2 ) (degG1 +G2 (u))3 :
  
(degG1 +G2 (u))3 = (degG1 (u) + p2 )3 + (degG2 (u) + p1 )3
u∈V (G1 +G2 ) u∈V (G1 ) u∈V (G2 )

= 3p2 M1 (G1 ) + 3p1 M1 (G2 ) + 6(p21 q2 + p22 q1 )


+ p1 p2 (p21 + p22 ) + (degG1 (u))3
u∈V (G1 )

+ (degG2 (u))3 .
u∈V (G2 )

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Similarly, we have

Theorem 4.3. For any two graphs G1 and G2 with |V (G1 )| = p1 , |V (G2 )| = p2 and
|E(G1 )| = q1 , |E(G2 )| = q2 , the second entire Zagreb index of G1 + G2 is given by

M2E (G1 + G2 )
= M2E (G1 ) + M2E (G2 ) + 3[p2 M1E (G1 ) + p1 M1E (G2 )]
1 1
+ (13p22 + 4p1 p2 + 8q2 )M1 (G1 ) + (13p21 + 4p1 p2 + 8q1 )M1 (G2 )
2 2
1
+ p1 p2 [(p1 + p2 )3 − (p1 + p2 )(3(p1 + p2 ) − 4) − (p1 − 2)2 − (p2 − 2)2 ]
2
+ p2 q1 [4(p1 + p2 )2 + 2(p22 − p21 ) − 3(5p2 + 2p1 ) + 2(q1 + 6q2 + 2)]
+ p1 q2 [4(p1 + p2 )2 + 2(p21 − p22 ) − 3(5p1 + 2p2 ) + 2(q2 + 6q1 + 2)].

Proof. We have,

M2E (G1 + G2 ) = 4qG1 +G2 − 2M1E (G1 + G2 ) − 2M1 (G1 + G2 )

1  
+ (degG1 +G2 (u))4 + (degG1 +G2 (u))2
2
u∈V (G1 +G2 ) u∈V (G1 +G2 )

× degG1 +G2 (v) + M2 (G1 + G2 )
v∈NG1 +G2 (u)
 2
1  
+  degG1 +G2 (v)
2
u∈V (G1 +G2 ) v∈NG1 +G2 (u)

= 4(q1 + q2 + p1 p2 ) − 2M1E (G1 + G2 ) − 2M1 (G1 + G2 )

1 
+ M2 (G1 + G2 ) + (degG1 (u) + p2 )4
2
u∈V (G1 )

1  
+ (degG2 (u) + p1 )4 + (degG1 (u) + p2 )2
2
u∈V (G2 ) u∈V (G1 )
 
 
× (degG1 (v) + p2 ) + (degG2 (v) + p1 )
v∈NG1 (u) v∈V (G2 )

 
+ (degG2 (u) + p1 )2  (degG2 (v) + p1 )
u∈V (G2 ) v∈NG2 (u)


+ (degG1 (v) + p2 )
v∈V (G1 )

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Entire Zagreb indices of graphs

 2
1   
+  (degG1 (v) + p2 ) + (degG2 (v) + p1 )
2
u∈V (G1 ) v∈NG1 (u) v∈V (G2 )
 2
1   
+  (degG2 (v) + p1 ) + (degG1 (v) + p2 ) .
2
u∈V (G2 ) v∈NG2 (u) v∈V (G1 )

By using Proposition 4.1, Theorem 4.2 and easy computations, the result is
obtained.

The following are directly obtained from above:

Corollary 4.4. For the complete bipartite graph Kr,m ,

(1) M1E (Kr,m ) = rm[(r + m)2 − 3(r + m) + 4].
(2) M2E (Kr,m ) = 12 rm[(r + m)3 − (r + m)(3(r + m) − 4) − (r − 2)2 − (m − 2)2 ].

Proof. The proof is obtained by applying Theorems 4.2 and 4.3 on Kr,m = Kr +
Km .

Corollary 4.5. For the wheel graph Wp ,

(1) M1E (Wp ) = (p − 1)(p2 + p + 24).
(2) M2E (Wp ) = 12 (p − 1)(p3 + 26p + 92).

Proof. The proof is obtained by applying Theorems 4.2 and 4.3 on Wp = K1 +

Cp−1 .

Corona Product
The corona product G1 ◦ G2 of two graphs G1 and G2 , where |V (G1 )| = p1 ,
|V (G2 )| = p2 and |E(G1 )| = q1 , |E(G2 )| = q2 is the graph obtained by taking
|V (G1 )| copies of G2 and joining each vertex of the ith copy with vertex u ∈ V (G1 ).
Obviously, |V (G1 ◦ G2 )| = p1 (p2 + 1) and |E(G1 ◦ G2 )| = q1 + p1 (q2 + p2 ). It
follows from the definition of the corona product G1 ◦ G2 , the degree of each vertex
v ∈ G1 ◦ G2 is given by

degG1 (v) + p2 , if v ∈ V (G1 );
degG1 ◦G2 (v) =
degG2 (v) + 1, if v ∈ V (G2 ).
We recall the following well-known result to be used in Theorems 4.7 and 4.8:

Proposition 4.6. For any two graphs G1 and G2 with |V (G1 )| = p1 , |V (G2 )| = p2
and |E(G1 )| = q1 , |E(G2 )| = q2 , the first and second Zagreb indices of G1 ◦ G2 are
given by
(1) M1 (G1 ◦ G2 ) = M1 (G1 ) + p1 M1 (G2 ) + 4(p1 q2 + p2 q1 ) + p1 p2 (p2 + 1).

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A. Alwardi et al.

(2) M2 (G1 ◦ G2 ) = M2 (G1 ) + p1 M2 (G2 ) + p2 M1 (G1 ) + p1 M1 (G2 ) + p2 q1 (p2 + 2) +

q2 (p1 + 4q1 ) + p1 p2 (p2 + 2q2 ).
Now we give our main results for Corona product:
Theorem 4.7. For any two graphs G1 and G2 with |V (G1 )| = p1 , |V (G2 )| = p2
and |E(G1 )| = q1 , |E(G2 )| = q2 , the first entire Zagreb index of G1 ◦ G2 is given by
M1E (G1 ◦ G2 ) = M1E (G1 ) + p1 M1E (G2 ) + 5[p2 M1 (G1 ) + p1 M1 (G2 )]
+ p1 p2 (p22 − p2 + 4q2 + 2) + 8p2 q1 (p2 − 1) − 4q2 (p1 − 2q1 ).

Proof. We have,
M1E (G1 ◦ G2 ) = 4qG1 ◦G2 − 3M1 (G1 ◦ G2 ) + 2M2 (G1 ◦ G2 )

+ (degG1 ◦G2 (u))3 .
u∈V (G1 ◦G2 )

Since, qG1 ◦G2 = q1 + p1 (q2 + p2 ) and M1 (G1 ◦ G2 ), M2 (G1 ◦ G2 ) are obtained in


Proposition 4.6, it only remains to calculate u∈V (G1 ◦G2 ) (degG1 ◦G2 (u))3 .
  
(degG1 ◦G2 (u))3 = (degG1 (u) + p2 )3 + p1 (degG2 (u) + 1)3
u∈V (G1 ◦G2 ) u∈V (G1 ) u∈V (G2 )

= 3p2 M1 (G1 ) + 3p1 M1 (G2 ) + 6(p22 q1 + p1 q2 )


+ p1 p2 (p22 + 1) + (degG1 (u))3
u∈V (G1 )

+ p1 (degG2 (u))3 .
u∈V (G2 )

Theorem 4.8. For any two graphs G1 and G2 with |V (G1 )| = p1 , |V (G2 )| = p2
and |E(G1 )| = q1 , |E(G2 )| = q2 , the second entire Zagreb index of G1 ◦ G2 is
given by
M2E (G1 ◦ G2 ) = M2E (G1 ) + p1 M2E (G2 ) + 3[p2 M1E (G1 ) + p1 M1E (G2 )]
1 1
+ (13p22 + 5p2 + 8q2 )M1 (G1 ) + (13p1 + 4p1 p2 + 8q1 )M1 (G2 )
2 2
1
+ p1 p2 (p32 − p22 + 5p2 + 4q2 (p2 + 1) − 3) + p1 q2 (2q2 − 5)
2
+ p2 q1 (6p22 − 7p2 + 12q2 ).

Proof. We have,
M2E (G1 ◦ G2 ) = 4qG1 ◦G2 − 2M1E (G1 ◦ G2 ) − 2M1 (G1 ◦ G2 )
1  
+ (degG1 ◦G2 (u))4 + (degG1 ◦G2 (u))2
2
u∈V (G1 ◦G2 ) u∈V (G1 ◦G2 )

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Entire Zagreb indices of graphs


× degG1 ◦G2 (v) + M2 (G1 ◦ G2 )
v∈NG1 ◦G2 (u)
 2
1  
+  degG1 ◦G2 (v)
2
u∈V (G1 ◦G2 ) v∈NG1 ◦G2 (u)

= 4(q1 + p1 q2 + p1 p2 ) − 2M1E (G1 ◦ G2 ) − 2M1 (G1 ◦ G2 )

1 
+ M2 (G1 ◦ G2 ) + (degG1 (u) + p2 )4
2
u∈V (G1 )

1  
+ p1 (degG2 (u) + 1)4 + (degG1 (u) + p2 )2
2
u∈V (G2 ) u∈V (G1 )
 
 
× (degG1 (v) + p2 ) + (degG2 (v) + 1)
v∈NG1 (u) v∈V (G2 )

 
+ (degG2 (u) + 1)2 p1 (degG2 (v) + 1)
u∈V (G2 ) v∈NG2 (u)
 
 1  
+ (degG1 (v) + p2 ) +  (degG1 (v) + p2 )
2
v∈V (G1 ) u∈V (G1 ) v∈NG1 (u)
2
 p1
1 
+ (degG2 (v) + 1) +
2 i=1
v∈V (G2 ) u∈V (G2 )
 2

× (degG2 (v) + 1) + degG1 (wi ) + p2  .
v∈NG2 (u)

By using Proposition 4.6, Theorem 4.7 and simple computations the result is
obtained.
We now give two applications of this result by calculating the first and second
entire Zagreb indices of the Corona product of two cycle graphs and one cycle
graph together with one path graph. As the calculations are straightforward, we
omit them:

Example 4.9. For any cycle Cp1 and any path Pp2 with p2 ≥ 3,

(1) M1E (Cp1 ◦ Pp2 ) = p1 (p32 + 11p22 + 42p2 − 42).


780p1, if p2 = 3;
(2) M2E (Cp1 ◦ Pp2 ) = 1
 p1 (p4 + 15p3 + 87p2 + 191p2 − 284), otherwise.
2 2 2
2
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A. Alwardi et al.

Example 4.10. For any two cycles Cp1 and Cp2 ,

(1) M1E (Cp1 ◦ Cp2 ) = p1 (p32 + 11p22 + 46p2 + 8).
(2) M2E (Cp1 ◦ Cp2 ) = 12 p1 (p42 + 15p32 + 91p22 + 251p2 + 32).

Acknowledgment
Ismail Naci Cangul was supported by Uludag University Research Fund, Project
No. F-2015/17.

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