Math. Sci. Lett. 6, No. 2, 193-198 (2017)
193
Mathematical Sciences Letters An International Journal http://dx.doi.org/10.18576/msl/060212
The Hyper-Zagreb Index of Four Operations on Graphs B. Basavanagoud∗ and Shreekant Patil Department of Mathematics, Karnatak University, Dharwad - 580 003, Karnataka, India Received: 27 May 2016, Revised: 21 Nov. 2016, Accepted: 23 Nov. 2016 Published online: 1 May 2017
Abstract: The hyper-Zagreb index of a connected graph G, denoted by HM(G), is defined as HM(G) =
∑
[dG (u) + dG (v)]2
uv∈E(G)
where dG (z) is the degree of a vertex z in G. In this paper, we study the hyper-Zagreb index of four operations on graphs. Keywords: Degree, Zagreb index, hyper-Zagreb index, operation on graphs
1 Introduction
Shirdel et al.[15] introduced a new Zagreb index of a graph G named hyper-Zagreb index and is defined as:
In this paper, we are concerned with simple connected graphs. Let G be such a graph with vertex set V (G), |V (G)| = n, and edge set E(G), |E(G)| = m. As usual, n is order and m is size of G. If u and v are two adjacent vertices of G, then the edge connecting them will be denoted by uv. The degree of a vertex w ∈ V (G) is the number of vertices adjacent to w and is denoted by dG (w). Line graph L = L(G); V (L) = E(G) and two vertices of L are adjacent if the corresponding edges of G are incident with a common vertex. We refer to [11] for unexplained terminology and notation. A graphical invariant is a number related to a graph, in other words, it is a fixed number under graph automorphisms. In chemical graph theory, these invariants are also called the topological indices. The first and second Zagreb indices are defined as M1 (G) =
∑ dG (u)2 and M2 (G) =
u∈V (G)
∑
HM(G) =
∑
(dG (u) + dG(v))2 .
uv∈E(G)
Let G be a graph with vertex set V (G) and edge set E(G), there are four related graphs as follows: • Subdivision graph S = S(G) [11]; V (S) = V (G) ∪ E(G) and the vertex of S corresponding to the edge uv of G is inserted in the edge uv of G; • Semitotal-point graph T2 = T2 (G) [14]; V (T2 ) = V (G) ∪ E(G) and E(T2 ) = E(S) ∪ E(G); • Semitotal-line graph T1 = T1 (G) [14]; V (T1 ) = V (G) ∪ E(G) and E(T1 ) = E(S) ∪ E(L); • Total graph T = T (G) [4]; V (T ) = V (G) ∪ E(G) and E(T ) = E(S) ∪ E(G) ∪ E(L).
dG (u)dG (v)
uv∈E(G)
respectively. The first Zagreb index can also expressed as [7] M1 (G) =
∑
[dG (u) + dG (v)].
uv∈E(G)
The vertex-degree-based graph invariant F(G) =
∑ dG (v)3 =
v∈V (G)
∑
[dG (u)2 + dG (v)2 ]
uv∈E(G)
was encountered in [10]. Recently there has been some interest to F, called “forgotten topological index”[9]. ∗ Corresponding
Figure 1: Graph G and its S(G), T2 (G), T1 (G) and T (G). In Fig. 1. The vertices of transformation graphs S(G), T2 (G), T1 (G), T (G) corresponding to the vertices of the parent graph G, are indicated by circles. The vertices of
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B. Basavanagoud, Shreekant Patil: The hyper-Zagreb index of four...
these graphs corresponding to the edges of the parent graph G are indicated by squares. For a given graph Gi , its vertex and edge sets will be denoted by V (Gi ) and E(Gi ), respectively, and their cardinalities by ni and mi , respectively, where i = 1, 2. The Cartesian product G1 × G2 of graphs G1 and G2 has the vertex set V (G1 × G2 ) = V (G1 ) × V (G2 ) and (u1 , v1 )(u2 , v2 ) is an edge of G1 × G2 if and only if [u1 = u2 and v1 v2 ∈ E(G2 )] or [v1 = v2 and u1 u2 ∈ E(G1 )]. In the recent paper [8], Eliasi and Taeri introduced four new operations on graphs as follows: Let F ∈ {S, T2, T1 , T }. The F-sum of G1 and G2 , denoted by G1 +F G2 , is a graph with the set of vertices V (G1 +F G2 ) = (V (G1 ) ∪ E(G1 )) × V (G2 ) and two vertices (u1 , u2 ) and (v1 , v2 ) of G1 +F G2 are adjacent if and only if [u1 = v1 ∈ V (G1 ) and u2 v2 ∈ E(G2 )] or [u2 = v2 ∈ V (G2 ) and u1 v1 ∈ E(F(G1 ))]. Thus, they obtained four new operations as G1 +S G2 , G1 +T2 G2 , G1 +T1 G2 and G1 +T G2 and studied the Wiener indices of these graphs. In [6], Deng et al. gave the expressions for first and second Zagreb indices of these new graphs.
Proof. By definition of hyper-Zagreb index, we have HM(G1 +S G2 )
∑
=
[dG1 +S G2 (u1 , v1 ) + dG1 +S G2 (u2 , v2 )]2
(u1 ,v1 )(u2 ,v2 )∈E(G1 +S G2 )
∑
=
∑
[dG1 +S G2 (u, v1 ) + dG1 +S G2 (u, v2 )]2
u∈V (G1 ) v1 v2 ∈E(G2 )
∑
+
∑
[dG1 +S G2 (u1 , v) + dG1 +S G2 (u2 , v)]2
v∈V (G2 ) u1 u2 ∈E(S(G1 ))
= A1 + A2
where A1 , A2 are the sums of the above terms, in order. A1 =
∑
∑
[dG1 (u) + dG2 (v1 ) + dG1 (u) + dG2 (v2 )]2
∑
∑
2 [4dG (u) + (dG2 (v1 ) + dG2 (v2 ))2 1
u∈V (G1 ) v1 v2 ∈E(G2 )
=
u∈V (G1 ) v1 v2 ∈E(G2 )
+ 4dG1 (u)(dG2 (v1 ) + dG2 (v2 ))] = 4m2 M1 (G1 ) + n1 HM(G2 ) + 8m1 M1 (G2 ). A2 =
∑
∑
[dS(G1 ) (u1 ) + dS(G1 ) (u2 ) + dG2 (v)]2
∑
∑
2 [(dS(G1 ) (u1 ) + dS(G1 ) (u2 ))2 + dG (v) 2
v∈V (G2 ) u1 u2 ∈E(S(G1 ))
=
v∈V (G2 ) u1 u2 ∈E(S(G1 ))
+ 2dG2 (v)(dS(G1 ) (u1 ) + dS(G1 ) (u2 ))] = n2 HM(S(G1 )) + 2m1 M1 (G2 ) + 4m2 M1 (S(G1 )).
Note that M1 (S(G1 )) = M1 (G1 ) + 4m1 and HM(S(G1 )) = 4M1 (G1 ) + F(G1 ) + 8m1. ∴ A2 = (4n2 + 4m2)M1 (G1 ) + 2m1M1 (G2 ) + n2F(G1 ) + 8n2 m1 + 16m1m2 . Adding A1 and A2 , we get the desired result. Theorem 2.2. Let G1 and G2 be the graphs. Then HM(G1 +T2 G2 ) = 8(5m2 + n2 )M1 (G1 ) + 22m1M1 (G2 ) +n1 HM(G2 ) + 4n2 HM(G2 ) + 4n2 F(G1 ) + 8m1 n2 + 16m1 m2 . Proof. By definition of hyper-Zagreb index, we have HM(G1 +T2 G2 )
∑
=
(u1 ,v1 )(u2 ,v2 )∈E(G1 +T2 G2 )
=
Figure 2: Graphs G1 and G2 and G1 +F G2 .
∑
∑
[dG1 +T2 G2 (u1 , v) + dG1 +T2 G2 (u2 , v)]2 .
Note that E(T2 (G1 )) = E(G1 ) ∪ E(S(G1)). HM(G1 +T2 G2 ) =
∑
∑
[dG1 +T2 G2 (u, v1 ) + dG1 +T2 G2 (u, v2 )]2
u∈V (G1 ) v1 v2 ∈E(G2 )
∑
∑
[dG1 +T2 G2 (u1 , v) + dG1 +T2 G2 (u2 , v)]2
v∈V (G2 ) u1 u2 ∈E(T2 (G1 )) u1 ,u2 ∈V (G1 )
2 Main results +
∑
v∈V (G2 )
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[dG1 +T2 G2 (u, v1 ) + dG1 +T2 G2 (u, v2 )]2
v∈V (G2 ) u1 u2 ∈E(T2 (G1 ))
+
Theorem 2.1. Let G1 and G2 be the graphs. Then HM(G1 +S G2 ) = 4(n2 + 2m2 )M1 (G1 ) + 10m1M1 (G2 ) + n1 HM(G2 ) + n2F(G1 ) + 8n2m1 + 16m1m2 .
∑
u∈V (G1 ) v1 v2 ∈E(G2 )
+
In this paper, we study the hyper-Zagreb index of G1 +S G2 , G1 +T2 G2 , G1 +T1 G2 and G1 +T G2 . Readers interested in more information on computing topological indices of graph operations can be referred to [1,2,3,5, 12,13].
∑
[dG1 +T2 G2 (u1 , v1 ) + dG1 +T2 G2 (u2 , v2 )]2
∑
u1 u2 ∈E(T2 (G1 )) u1 ∈V (G1 ),u2 ∈V (T2 (G1 ))\V (G1 )
+ dG1 +T2 G2 (u2 , v)]2 = B1 + B2 + B3
[dG1 +T2 G2 (u1 , v)
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where B1 , B2 and B3 are the sums of the above terms, in order. B1 =
∑
∑
[2dT2 (G1 ) (u) + dG2 (v1 ) + dG2 (v2 )]2
∑
∑
[4dG1 (u) + dG2 (v1 ) + dG2 (v2 )]2
∑
∑
[16dG2 1 (u) + (dG2 (v1 ) + dG2 (v2 ))2
u∈V (G1 ) v1 v2 ∈E(G2 )
=
u∈V (G1 ) v1 v2 ∈E(G2 )
=
Partition the edge set E(T1 (G1 )) into E(S(G1 )) and E(L(G1 )). HM(G1 +T1 G2 ) u∈V (G1 ) v1 v2 ∈E(G2 )
∑
+
v∈V (G2 )
∑
∑
+
[2dG2 (v) + dT2 (G1 ) (u1 )
=
∑
∑
[2dG2 (v) + 2dG1 (u1 ) + 2dG1 (u2 )]2
∑
∑
[4dG2 2 (v) + 4(dG1 (u1 ) + dG1 (u2 ))2
C1 =
v∈V (G2 )
[dG1 +T1 G2 (u1 , v)
∑
∑
[2dT1 (G1 ) (u) + dG2 (v1 ) + dG2 (v2 )]2
∑
∑
[2dG1 (u) + dG2 (v1 ) + dG2 (v2 )]2
∑
∑
[4dG2 1 (u) + (dG2 (v1 ) + dG2 (v2 ))2
u∈V (G1 ) v1 v2 ∈E(G2 )
=
+ 8dG2 (v)(dG1 (u1 ) + dG1 (u2 ))] = 4m1 M1 (G2 ) + 4n2HM(G1 ) + 16m2M1 (G1 ).
∑
∑
u1 u2 ∈E(T1 (G1 )) u1 ,u2 ∈V (T1 (G1 ))\V (G1 )
where C1 , C2 and C3 are the sums of the above terms, in order.
v∈V (G2 ) u1 u2 ∈E(G1 )
B3 =
[dG1 +T1 G2 (u1 , v)
= C1 + C2 + C3
2
v∈V (G2 ) u1 u2 ∈E(G1 )
=
∑
u1 u2 ∈E(T1 (G1 )) u1 ∈V (G1 ),u2 ∈V (T1 (G1 ))\V (G1 )
+ dG1 +T1 G2 (u2 , v)]2
v∈V (G2 ) u1 u2 ∈E(T2 (G1 )) u1 ,u2 ∈V (G1 )
+ dT2 (G1 ) (u2 )]
[dG1 +T1 G2 (u, v1 ) + dG1 +T1 G2 (u, v2 )]2
+ dG1 +T1 G2 (u2 , v)]2 v∈V (G2 )
∑
∑
∑
=
u∈V (G1 ) v1 v2 ∈E(G2 )
+ 8dG1 (u)(dG2 (v1 ) + dG2 (v2 ))] = 16m2 M1 (G1 ) + n1HM(G2 ) + 16m1M1 (G2 ). B2 =
195
u∈V (G1 ) v1 v2 ∈E(G2 )
=
∑
[dT2 (G1 ) (u1 ) + dG2 (v)
∑
[2dG1 (u1 ) + dG2 (v) + 2]2 .
u∈V (G1 ) v1 v2 ∈E(G2 )
+ 4dG1 (u)(dG2 (v1 ) + dG2 (v2 ))] = 4m2 M1 (G1 ) + n1 HM(G2 ) + 8m1M1 (G2 ).
u1 u2 ∈E(T2 (G1 )) u1 ∈V (G1 ),u2 ∈V (T2 (G1 ))\V (G1 )
+ dT2 (G1 ) (u2 )]2
∑
=
v∈V (G2 )
u1 u2 ∈E(T2 (G1 )) u1 ∈V (G1 ),u2 ∈V (T2 (G1 ))\V (G1 )
C2 =
The quantity 2dG1 (u1 )+ dG2 (v)+ 2 appears dG1 (u1 ) times. Hence,
∑
B3 =
∑
∑
v∈V (G2 )
+ dT1 (G1 ) (u2 )]2
dG1 (u)[4dG2 1 (u) + dG2 2 (v) + 4
=
∑
v∈V (G2 )
v∈V (G2 ) u∈V (G1 )
+ 4dG2 (v) + 4dG1 (u)(dG2 (v) + 2)] = 4n2 F(G1 ) + 2m1 M1 (G2 ) + 8m1n2 + 16m1m2 + 8m2 M1 (G1 ) + 8n2M1 (G1 ). Adding B1 , B2 and B3 , we get the desired result. Theorem 2.3. Let G1 and G2 be the graphs. Then HM(G1 +T1 G2 ) = 16m2 M1 (G1 ) + 10m1M1 (G2 ) + n1 HM(G2 ) + 4n2 HM(G1 ) + n2 F(G1 ) + n2 [ [dG1 (wi ) + 2dG1 (w j ) + dG1 (wk )]2 ]. ∑ wi w j ∈E(G1 ),w j wk ∈E(G1 )
∑
[dT1 (G1 ) (u1 ) + dG2 (v)
∑
[dG1 (u1 ) + dG2 (v)
u1 u2 ∈E(T1 (G1 )) u1 ∈V (G1 ),u2 ∈V (T1 (G1 ))\V (G1 )
u1 u2 ∈E(T1 (G1 )) u1 ∈V (G1 ),u2 ∈V (T1 (G1 ))\V (G1 )
+ dT1 (G1 ) (u2 )]2
=
∑
v∈V (G2 )
∑
[dG2 1 (u1 ) + dG2 2 (v)
u1 u2 ∈E(T1 (G1 )) u1 ∈V (G1 ),u2 ∈V (T1 (G1 ))\V (G1 )
+ 2dG1 (u1 )dG2 (v) + dT21 (G1 ) (u2 ) + 2dT1 (G1 ) (u2 )(dG1 (u1 ) + dG2 (v))]
Proof. By definition of hyper-Zagreb index, we have HM(G1 +T1 G2 )
∑
=
(u1 ,v1 )(u2 ,v2 )∈E(G1 +T1 G2 )
=
∑
∑
u∈V (G1 ) v1 v2 ∈E(G2 )
+
∑
∑
[dG1 +T1 G2 (u1 , v1 ) + dG1 +T1 G2 (u2 , v2 )]2
[dG1 +T1 G2 (u, v1 ) + dG1 +T1 G2 (u, v2 )]
2
[dG1 +T1 G2 (u1 , v) + dG1 +T1 G2 (u2 , v)]2 .
v∈V (G2 ) u1 u2 ∈E(T1 (G1 ))
=
∑
∑
dG1 (u)[dG2 1 (u) + dG2 2 (v) + 2dG1 (u)dG2 (v)]
v∈V (G2 ) u∈V (G1 )
+
∑
v∈V (G2 )
∑
[dT21 (G1 ) (u2 )
u1 u2 ∈E(T1 (G1 )) u1 ∈V (G1 ),u2 ∈V (T1 (G1 ))\V (G1 )
+ 2dT1 (G1 ) (u2 )(dG1 (u1 ) + dG2 (v))].
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Note that for u2 ∈ V (T1 (G1 ))\V (G1 ), dT1 (G1 ) (u2 ) = dG1 (u) + dG1 (w) where u2 = uw ∈ E(G1 ). C2 = n2 F(G1 ) + 2m1 M1 (G2 ) + 4m2 M1 (G1 )
∑
+
v∈V (G2 )
∑
where D1 , D2 , D3 and D4 are the sums of the above terms, in order. D1 =
[(dG1 (u)
=
+ dG1 +T1 G2 (u2 , v)]
∑
=
v∈V (G2 )
[dG1 +T1 G2 (u1 , v)
v∈V (G2 )
Adding C1 , C2 and C3 , we get the desired result. Theorem 2.4. Let G1 and G2 be the graphs. Then HM(G1 +T G2 ) = 48m2 M1 (G1 ) + 22m1 M1 (G2 ) + 10n2HM(G1 ) + n1 HM(G2 ) + 4n2 F(G1 ) + n2 [ [dG1 (wi ) + 2dG1 (w j ) + dG1 (wk )]2 ]. ∑
[dT (G1 ) (u1 ) + dG2 (v)
∑
[2dG1 (u1 ) + dG2 (v)
u1 u2 ∈E(T (G1 )) u1 ∈V (G1 ),u2 ∈V (T (G1 ))\V (G1 )
v∈V (G2 )
∑
[4dG2 1 (u1 ) + dG2 2 (v)
u1 u2 ∈E(T (G1 )) u1 ∈V (G1 ),u2 ∈V (T (G1 ))\V (G1 )
2 + 4dG1 (u1 )dG2 (v) + dT(G (u2 ) + 2dT(G1 ) (u2 )(2dG1 (u1 ) 1)
HM(G1 +T G2 ) 2
+ dG2 (v))]
(u1 ,v1 )(u2 ,v2 )∈E(G1 +T G2 )
∑
∑
=
Proof. By definition of hyper-Zagreb index, we have [dG1 +T G2 (u1 , v1 ) + dG1 +T G2 (u2 , v2 )]
∑
u1 u2 ∈E(T (G1 )) u1 ∈V (G1 ),u2 ∈V (T (G1 ))\V (G1 )
+ dT (G1 ) (u2 )]2
wi w j ∈E(G1 ),w j wk ∈E(G1 )
∑
∑
=
v∈V (G2 )
∑
[2dG2 (v) + 2dG1 (u1 ) + 2dG1 (u2 )]2
+ dT (G1 ) (u2 )]2
[dG1 (wi ) + dG1 (w j ) + dG1 (w j )
+ dG1 (wk )]2 ].
=
∑
D3 =
[dT1 (G1 ) (u1 ) + dT1 (G1 ) (u2 )]2
wi w j ∈E(G1 ),w j wk ∈E(G1 )
=
∑
= 4m1 M1 (G2 ) + 4n2 HM(G1 ) + 16m2 M1 (G1 ).
2
∑
[2dG2 (v) + dT (G1 ) (u1 ) + dT (G1 ) (u2 )]2
v∈V (G2 ) u1 u2 ∈E(G1 )
u1 u2 ∈E(T1 (G1 )) u1 ,u2 ∈V (T1 (G1 ))\V (G1 )
= n2 [
∑
∑
=
∑
[4dG1 (u) + dG2 (v1 ) + dG2 (v2 )]2
v∈V (G2 ) u1 u2 ∈E(T (G1 )) u1 ,u2 ∈V (G1 )
= n2 F(G1 ) + 2m1 M1 (G2 ) + 4n2 HM(G1 ) + 12m2 M1 (G1 ).
∑
∑
∑
D2 =
+ 2n2 [F(G1 ) + 2M2 (G1 )] + 8m2 M1 (G1 )
u1 u2 ∈E(T1 (G1 )) u1 ,u2 ∈V (T1 (G1 ))\V (G1 )
∑
u∈V (G1 ) v1 v2 ∈E(G2 )
+ 2(dG1 (u) + dG1 (w))dG2 (v)]
∑
[2dT (G1 ) (u) + dG2 (v1 ) + dG2 (v2 )]2
= 16m2 M1 (G1 ) + n1HM(G2 ) + 16m1M1 (G2 ).
= n2 F(G1 ) + 2m1 M1 (G2 ) + 4m2 M1 (G1 ) + 2n2 HM(G1 )
v∈V (G2 )
∑
u∈V (G1 ) v1 v2 ∈E(G2 )
u1 u2 ∈E(T1 (G1 )) u1 ∈V (G1 ),u2 ∈V (T1 (G1 ))\V (G1 )
+ dG1 (w))2 + 2(dG1 (u) + dG1 (w))dG1 (u1 )
C3 =
∑
[dG1 +T G2 (u, v1 ) + dG1 +T G2 (u, v2 )]2
u∈V (G1 ) v1 v2 ∈E(G2 )
+
∑
∑
[dG1 +T G2 (u1 , v) + dG1 +T G2 (u2 , v)]2 .
∑
∑
[dG1 +T G2 (u, v1 ) + dG1 +T G2 (u, v2 )]
∑
∑
v∈V (G2 )
2
[dG1 +T G2 (u1 , v) + dG1 +T G2 (u2 , v)]
v∈V (G2 ) u1 u2 ∈E(T (G1 )) u1 ,u2 ∈V (G1 )
+
∑
v∈V (G2 )
∑
[dG1 +T G2 (u1 , v)
u1 u2 ∈E(T (G1 )) u1 ∈V (G1 ),u2 ∈V (T (G1 ))\V (G1 )
+ dG1 +T G2 (u2 , v)]2 +
∑
v∈V (G2 )
∑
u1 u2 ∈E(T1 (G1 )) u1 ,u2 ∈V (T1 (G1 ))\V (G1 )
+ dG1 +T1 G2 (u2 , v)]2 = D1 + D2 + D3 + D4
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[dG1 +T1 G2 (u1 , v)
∑
+
dG1 (u)[4dG2 1 (u) + dG2 2 (v) + 4dG1 (u)dG2 (v)]
∑
u1 u2 ∈E(T (G1 )) u1 ∈V (G1 ),u2 ∈V (T (G1 ))\V (G1 )
[dT2 (G1 ) (u2 )
+ 2dT (G1 ) (u2 )(2dG1 (u1 ) + dG2 (v))].
u∈V (G1 ) v1 v2 ∈E(G2 )
+
∑
v∈V (G2 ) u∈V (G1 )
Note that E(T (G1 )) = E(G1 ) ∪ E(S(G1)) ∪ E(L(G1 )). HM(G1 +T G2 ) =
∑
=
v∈V (G2 ) u1 u2 ∈E(T (G1 ))
2
Note that for u2 ∈ V (T (G1 ))\V (G1 ), dT (G1 ) (u2 ) = dG1 (u) + dG1 (w) where u2 = uw ∈ E(G1 ). D3 = 4n2 F(G1 ) + 2m1 M1 (G2 ) + 8m2 M1 (G1 ) +
∑
v∈V (G2 )
∑
[(dG1 (u) + dG1 (w))2
u1 u2 ∈E(T (G1 )) u1 ∈V (G1 ),u2 ∈V (T (G1 ))\V (G1 )
+ 4(dG1 (u) + dG1 (w))dG1 (u1 ) + 2(dG1 (u) + dG1 (w))dG2 (v)] = 4n2 F(G1 ) + 2m1 M1 (G2 ) + 8m2 M1 (G1 ) + 2n2 HM(G1 ) + 4n2 [F(G1 ) + 2M2 (G1 )] + 8m2 M1 (G1 ) = 4n2 F(G1 ) + 2m1 M1 (G2 ) + 16m2 M1 (G1 ) + 6n2 HM(G1 ).
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D4 =
∑
v∈V (G2 )
∑
[dG1 +T G2 (u1 , v)
u1 u2 ∈E(T (G1 )) u1 ,u2 ∈V (T (G1 ))\V (G1 )
+ dG1 +T G2 (u2 , v)]2 =
∑
v∈V (G2 )
∑
[dT (G1 ) (u1 ) + dT(G1 ) (u2 )]2
u1 u2 ∈E(T (G1 )) u1 ,u2 ∈V (T (G1 ))\V (G1 )
= n2 [
∑
[dG1 (wi ) + dG1 (w j )
wi w j ∈E(G1 ),w j wk ∈E(G1 )
+ dG1 (w j ) + dG1 (wk )]2 ]. Adding D1 , D2 , D3 and D4 , we get the desired result. Applying the above four theorems to the graphs G1 = Pr and G2 = Pq , we have (i)HM(Pr +S Pq ) = 136rq − 138r − 150q + 124, q > 2; (ii)HM(Pr +T2 Pq ) = 416rq − 338r − 576q + 388, r, q > 2; (iii)HM(P r +T1 Pq ) 192rq − 154r − 234q + 156 i f r = 3, q > 2; = 256rq − 154r − 428q + 156 i f r > 3, q > 2; (iv)HM(P r +T Pq ) 488rq − 354r − 696q + 420 i f r = 3, q > 2; = 552rq − 354r − 890q + 420 i f r > 3, q > 2.
197
[6] H. Deng, D. Sarala, S. K. Ayyaswamy, S. Balachandran, The Zagreb indices of four operations on graphs, Appl. Math. Comput. 275 (2016) 422–431. [7] T. Dosli ˇ c, ´ B. Furtula, A. Graovac, I. Gutman, S. Moradi, Z. Yarahmadi, On vertex-degree-based molecular structure descriptors, MATCH Commun. Math. Comput. Chem. 66 (2011) 613–626. [8] M. Eliasi, B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Appl. Math. 157 (2009) 794–803. [9] F. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015) 1184–1190. [10] I. Gutman, N. Trinajstic, ´ Graph theory and molecular orbitals. Total π -electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538. [11] F. Harary, Graph Theory, Addison-Wesley, Reading, Mass 1969. [12] M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math. 157 (2009) 804–811. [13] M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, The hyperWiener index of graph operations, Comput. Math. Appl. 56 (2008) 1402–1407. [14] E. Sampathkumar, S. B. Chikkodimath, Semitotal graphs of a graph-I, J. Karnatak Univ Sci. 18 (1973) 274–280. [15] G. H. Shirdel, H. Rezapour, A.M. Sayadi, The hyper-Zagreb index of graph operations, Iranian J. Math. Chem. 4(2) (2013) 213–220.
3 Conclusion In this paper, we have studied the hyper Zagreb index of new four sums of graphs. Also we apply our results to compute the hyper Zagreb index of Pr +S Pq , Pr +T2 Pq , Pr +T1 Pq and Pr +T Pq . For further research, one can study the other topological indices of these new operations.
Acknowledgement The first author acknowledges the financial support by UGC-SAP DRS-III, New Delhi, India for 2016-2021: F.510/3/DRS-III/2016(SAP-I) Dated: 29th Feb. 2016.
References [1] A. R. Ashrafi, T. Dosli ˇ c, ´ A. Hamzeh, The Zagreb coindices of graph operations, Discrete Appl. Math. 158 (2010) 1571– 1578. [2] B. Basavanagoud, S. Patil, A note on hyper-Zagreb index of graph operations, Iranian J. Math. Chem. 7(1) (2016) 89–92. [3] B. Basavanagoud, S. Patil, A note on hyper-Zagreb coindex of graph operations, J. Appl. Math. Comput. (2016) doi:10.1007/s12190-016-0986-y. [4] M. Behzad, A criterion for the planarity of a total graph, Pro. Cambridge Philos. Soc. 63 (1967) 697–681. [5] K. C. Das, A. Yurttas, M. Togan, A. S. Cevik, I. N. Cangul, The multiplicative Zagreb indices of graph operations, J. Inequal. Appl. 90 (2013) 1–14.
B. Basavanagoud received his both MSc and PhD degree in Mathematics from Gulbarga University, Gulbarga, Karnataka state, India. Currently he is working as Professor and Chairman, Department of Mathematics, Karnatak University, Dharwad, Karnataka state, India. He has published many research papers in reputed national and international journals. His area of interest lies mainly in Graph Theory. He is also life member of The Indian Mathematical Society, The Indian Science Congress Association and International Journal of Mathematical Sciences and Engineering Applications. He chaired the session in the International Congress of Mathematicians(ICM-2010) and delivered a contributed talk at the same.
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B. Basavanagoud, Shreekant Patil: The hyper-Zagreb index of four...
Shreekant Patil received his MSc degree in Mathematics from Karnatak University, Dharwad, Karnataka state, India. He is currently a research student in the Department of Mathematics, Karnatak University, Dharwad working under the guidance of Dr. B. Basavanagoud.
c 2017 NSP
Natural Sciences Publishing Cor.
