SOME VERTEX-DEGREE-BASED TOPOLOGICAL INDICES UNDER EDGE CORONA PRODUCT

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1 ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N SOME VERTEX-DEGREE-BASED TOPOLOGICAL INDICES UNDER EDGE CORONA PRODUCT I. Rezaee Abdolhosseinzadeh F. Rahbarnia M. Tavaoli Department of Applied Mathematics Ferdowsi University of Mashhad P. O. Box 59, Mashhad 9775, I.R. Iran ir rezaee899@um.ac.ir rahbarnia@um.ac.ir m tavaoli@um.ac.ir A. R. Ashrafi Department of Pure Mathematics Faculty of Mathematical Sciences University of Kashan, P. O. Box , Kashan I. R. Iran ashrafi@ashanu.ac.ir Abstract. A topological index is called vertex-degree-based if it can be defined by vertex degrees. The harmonic, atom-bond connectivity and Randić indices are three important examples of such topological indices. The aim of this paper is to find lower and upper bounds for Randić, harmonic and atom-bond connectivity indices of edge corona product of graphs. Some closed formulas are obtained when the factors are regular graphs. Keywords: Edge corona product, Randić index, harmonic index, atom-bond connectivity index.. Introduction and preliminaries Suppose G is a simple graph with vertex set V G and edge set EG. The Randić index of G, RG, is defined as the sum of over all edges degvi degv j uv EG, where degx, as a short dx, denotes the degree of a vertex x in G [0]. This parameter, sometimes referred to as connectivity index, has been used to characterize the degree of branching of organic compounds. As an example, this number successfully explained the occurrence of critical alloy compositions in 8 different binary alloys [9]. The higher order Randić indices are also of interest in chemical graph theory. For h, the h-th order Randić. Corresponding author

2 I. REZAEE ABDOLHOSSEINZADEH, F. RAHBARNIA, M. TAVAKOLI, A. R. ASHRAFI 8 index R h G of G is the sum of the term overall paths degv i degv i...degv ih v i, v i,..., v ih of length h contained as a subgraph in G [5, 6]. The case that h is ordinary Randić index. We encourage the reader to consult [7, 8, 5] and references therein for more information on this topic. The Harmonic index of a graph G, HG, is defined as HG uv EG d G ud G v. As far as we now, this graph invariant first appeared in [3]. The atom-bond connectivity index ABC index for short was introduced by Ernesto Estrada et al. for studying the stability of alanes and the strain energy of cycloalanes []. This index can be defined as ABCG dudv uv EG du.dv. Shwetha Shetty [] obtained exact formulas for the Harary index of join, corona product, Cartesian product, composition and symmetric difference of graphs. Zhong [6] obtained the minimum and maximum values of the harmonic index for simple connected graphs and trees. He also characterized the corresponding extremal graphs. Xu [] established some relationships between harmonic, Randić and ABC indices of graphs. The edge corona product of two graphs G and H, G H, is a graph obtained by taing a copy of G and EG copies of H and joining each end vertex of i-th edge of G to every vertex in the i-th copy of H [, 4]. Following Yan et al. [3], the graph RG is obtained from G by adding a new vertex corresponding to each edge of G, then joining each new vertex to the end vertices of the corresponding edge. Another way to describe RG is to replace each edge of G by a triangle. It is clear that if G is a graph and H is a trivial graph, then G H RG. In [4], Yero et al. studied the Randić index of corona product of graphs. In this paper, we continue this wor by computing the Randić index of edge corona product of graphs. Throughout this paper our notation is standard. A -regular graph is a graph in which degree of each vertex is equal to. In a graph G with at least one cycle, the length of a longest cycle is called its circumference and the length of a shortest cycle its girth. Our other notions are standard and can be taen from the standard boos on graph theory.. Main results The aim of this section is computing the Randić, atom bond connectivity and harmonic indices of edge corona product of graphs. The following simple lemma is crucial in our results. Lemma.. By definition of edge corona we have: If x V G then d G G x d G xn, If x V G then d G G x d G x. Theorem.. Let G and G be graphs. Thus, we have m R G G n m m m n n,

3 SOME VERTEX-DEGREE-BASED TOPOLOGICAL INDICES Proof. Let A A A 3 R G G ab EG uv EG a V G,u V G m δ n m m δ m n δ δ n. dadbn m n. du dv m m. dan du m n n. By summation of A, A and A 3, the result can be proved. Corollary.3. For i {, }, if G i be δ i -regular graph of order n i, then R G G m δ n m m δ m n δ δ n. Theorem.4. Let G and G be graphs. Then R G G n δ n n n m δ n m n n δ n 3/ d G u i d G v i d G v i d G v i dg v i d G u i d G u i. dg u i R G G n δ n n n m n m n n δ δ n 3/ δ δ d G u i d G v i d G v i d G v i dg v i d G u i d G u i. dg u i Proof. Suppose V {v, v,..., v n } and V {u, u,..., u n } are the vertex set of G and G, respectively. For v V i, N Gi v is the set of all adjacent vertices to v in G i. Suppose ρ h G denotes the set of all paths of length h in G and Consider the following partition of paths of length two in G G.

4 I. REZAEE ABDOLHOSSEINZADEH, F. RAHBARNIA, M. TAVAKOLI, A. R. ASHRAFI 84. Paths u i v j u, i where u i, u V and v j V,. Paths u i v j v, j where v j, v V and u i V, 3. Paths v i u j u, j where u j, u V and v i V, 4. Paths u i v j u, where u i, u V two different copies of G and v j V, 5. Paths v i u j v, i where u j V, v i, v V and v i v EG, 6. Paths of length two belonging to G, 7. Paths of length two belonging to m copies of G. Define T i to be the set of all paths of type i, i 7. Therefore, R G G 7 i A i, where A A u i u i u i3 {P ρ G G P T } u i,u V,v j V n j dui du i du i3 dg u i d G u n d G v j n d G v j dg v j n. n n n. δ n, i u i u i u i3 {P ρ G G P T } n i v j,v V,u i V n n i δ m n n, A 3 dg u i d G u dui du i du i3 dg u i d G v j d G v n n dg u i. u i u i u i3 {P ρ G G P T 3 } j d G v j v N G v j dui du i du i3 dg v i n d G u j d G u u j,u V,v i V dg v j d G v

5 SOME VERTEX-DEGREE-BASED TOPOLOGICAL INDICES A 4 A 5 n i n d G v i dg v i n. n m δ n, u i u i u i3 {P ρ G G P T 4 } j u N G u j dg u j d G u dui du i du i3 dg u i n d G v j d G u u i V t,u V l l t,v j V n j n d G v j d G v j d G v j n. n i δ n, n n δ. u i u i u i3 {P ρ G G P T 5 } u j V,v i v EG n v i v EG m n n, A 6 A 7 u i u i u i3 {P ρ G G P T 6 } v i v j v ρ G n 3/ dg u i d G u dui du i du i3 dg v i d G v n d G u j n dg v i d G v. dg u j j dui du i du i3 dg v i d G v j d G v n 3 v i v j v ρ G n 3/ d G v i u i u i u i3 {P ρ G G P T 7 } u i u j u ρ G dg v i d G v j d G v d G v i d G v i, dg v i dui du i du i3 dg u i d G u j d G u

6 I. REZAEE ABDOLHOSSEINZADEH, F. RAHBARNIA, M. TAVAKOLI, A. R. ASHRAFI 86 d G u i Now a simple calculations will complete the proof. d G u i d G u i. dg u i Corollary.5. For i {, }, if G i be δ i -regular graph of order n i, then R G G n δ δ n m n n m n n δ δ δ n 3/ δ δ d G u i d G v i δ n d G v i d G v i dg v i d G u i d G u i. dg u i To prove our main result, we state an important result of [4]. Lemma.6 [4]. Let G V, E be a graph with girth gg. If δ and gg > h, then the number of paths of length h in G is bounded by δ h dudu ρ h G u V h dudu. u V In what follows, let N denote the empty graph of order. Theorem.7. Let G V, E be a graph with girth gg, minimum degree δ, and maximum degree. If δ and gg > h 3, then h R h G N δ h h 3 8 h δ h δ h R h G N h δ h 3 8 h h δ. dudu, u V h. dudu. u V h h δ Proof. The paths of length h in G contribute to R h G N in v i v i...v ih ρ h G h. h l d Gv il h

7 SOME VERTEX-DEGREE-BASED TOPOLOGICAL INDICES Moreover, each path of length h in G leads to paths of length h in G N ; thus, the paths of length h in G contribute to R h G N in v i v i...v ih ρ h G h h l dv i l h h l dv i l 4 h. h l dv i l Furthermore, each cycle of length h in G leads to h paths of length h in G N and also each cycle of length h in G leads to h paths of length h in G N ; thus, the cycle of length h and h in G contribute to R h G N in v i v i...v ih v i ζ h G v i v i...v ih v i ζ h G h h, h l dv i l h 4 h, h l dv i l respectively, where ζ h G denotes the set of cycles of length h contained as subgraphs in G. So, R h G N v i v i...v ih ρ h G h h v i v i...v ih v i ζ h G v i v i...v ih v i ζ h G ρ h G h δ h h h l dv i l h h l dv i l l dv i l 4 h h l dv i l h h h l dv i l h 4 h h l dv i l h δ h 4 h δ h h ζ h G h δ ζ h h G h 4 h δ. h By taing into account that ζ h G ρ h G, ζ h G ρ h G and Lemma.6 we obtain the upper bound and the lower bound.

8 I. REZAEE ABDOLHOSSEINZADEH, F. RAHBARNIA, M. TAVAKOLI, A. R. ASHRAFI 88 Theorem.8. Let G and G be graphs. Thus, we have HG G n HG m m m n n, HG G n HG m m δ m n δ n δ. Proof. The edges of G G are partitioned into three subsets E, E and E 3 as follows: E {e EG G e EG }, E {e EG G e EG i, i,,..., EG }, E 3 {e EG G e uv, u V G i, i,,..., EG, v V G }. Therefore, Where A A A 3 HG G uv E uv EG G du dv uv E d G G u d G G v A A A 3. n d G u n d G v n HG, uv E du dv m d G u d G v m m, uv E du dv d G u n dv m n n. uv E 3 uv E 3 By summation of A, A and A 3, the result can be proved. Also for the reverse bound we can do analogously. Corollary.9. For i {, }, if G i be δ i -regular graph of order n i, then HG G n HG m m δ m n δ n. Corollary.0. For a graph G of size m and an empty graph N of order, we have HG N HG m, HG N HG m δ.

9 SOME VERTEX-DEGREE-BASED TOPOLOGICAL INDICES Theorem.. Let G and G be graphs. Thus, we have m ABCG G n δ m m δ n δ n δ m n, n m ABCG G n m m δ n δ n m n. n δ δ Proof. The edges of G G are partitioned into three subsets E, E and E 3 as follows: E {e EG G e EG }, E {e EG G e EG i, i,,..., EG }, E 3 {e EG G e uv, u V G i, i,,..., EG, v V G }. Therefore, ABCG G Where A uv E A m A 3 uv E 3 uv EG G n d G u d G v n d G ud G v uv E d G G u d G G v A A A 3. d G G ud G G v m n δ, n d G u d G v m m δ, d G u d G v m n n d G u d G v n d G ud G v n δ δ. n By summation of A, A and A 3, the result can be proved. Also for the reverse bound we can do analogously. Corollary.. For i {, }, if G i be δ i -regular graph of order n i, then m ABCG G n δ m m δ δ n δ δ n δ m n. n δ δ

10 I. REZAEE ABDOLHOSSEINZADEH, F. RAHBARNIA, M. TAVAKOLI, A. R. ASHRAFI 90 Corollary.3. For a graph G of size m and an empty graph N of order, we have m δ ABCG N δ m, 4 m ABCG N m. δ 4 δ References [] K. P. Chithra, K. A. Germina and N. K. Sudev, On the Sparing Number of the Edge Corona of Graphs, Int. J. Comput. Appl., 8 05, -5. [] E. Estrada, L. Torres, L. Rodriguez and I. Gutman, An Atombond connectivity index: Modelling the enthalpy of formation of alanes, Indian J. Chem., , [3] S. Fajtlowicz, On conjectures of Graffiti. II, Congr., , [4] Y. Hou, W-C.Shiu, The spectrum of the edge corona of two graphs, Electron. J. Linear Algebra, 0 00, [5] L. B. Kier and L. H. Hall, Molecular Connectivity in Chemistry and Drug Research, Academic Press, New Yor, 976. [6] L. B. Kier and L. H. Hall, Molecular Connectivity in Structure Activity Analysis, Wiley, New Yor, 986. [7] X. Li and I. Gutman, Mathematical Aspects of Randić -Type Molecular Structure Descriptors, Faculty of Science, University of Kragujevac, 006. [8] X. Li and Y. Shi, A survey on the Randić index, MATCH Commun. Math. Comput. Chem., , [9] E. McCafferty, Introduction to corrosion science, Springer, New Yor, 009. [0] M. Randić, On characterization of molecular branching, J. Amer. Chem. Soc., , [] B. Shwetha Shetty, V. Loesha and P. S. Ranjini, On the harmonic index of graphs operations, Transactions on Combinatorics, 4 05, 5-4. [] X. Xu, Relationships between harmonic index and other topological indices, Appl. Math. Sci., 6 0,

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