On the Randić Index of Corona Product Graphs

Let G be a graph with vertex set V v1, v2, . . . , vn . Let δ vi be the degree of the vertex vi ∈ V . If the vertices vi1 , vi2 , . . . , vih 1 form a path of length h ≥ 1 in the graph G, then the hth order Randić index Rh of G is defined as the sum of the terms 1/ √ δ vi1 δ vi2 · · · δ vih 1 over all paths of length h contained as subgraphs in G. Lower and upper bounds for Rh, in terms of the vertex degree sequence of its factors, are obtained for corona product graphs. Moreover, closed formulas are obtained when the factors are regular graphs.


Introduction
In this work we consider simple graphs G V, E with n vertices and m edges.Let V v 1 , v 2 , . . ., v n be the vertex set of G.For every vertex v i ∈ V , δ v i represents the degree of the vertex v i in G.The maximum and minimum degree of the vertices of G will be denoted by Δ and δ, respectively.
The Randić index R 1 G of a graph G was introduced in 1975 1 and defined as This graph invariant, sometimes referred to as connectivity index, has been successfully related to a variety of physical, chemical, and pharmacological properties of organic molecules, and it has became into one of the most popular molecular-structure descriptors.
After the publication of the first paper 1 , mathematical properties of R 1 were extensively studied, see 2-6 and the references cited therein.
The higher-order Randić indices are also of interest in chemical graph theory.For h ≥ 1, the hth order Randić index R h G of a graph G is defined as where P h G denotes the set of paths of length h contained as subgraphs in G.Of the higherorder Randić indices the most frequently applied is R 2 7-10 .Estimations of the higher-order Randić index of regular graphs and semiregular bipartite graphs are given in 10 .In this paper we are interested in studying the higher-order Randić index, R h , for corona product graphs.Roughly speaking, we study the cases h 1, h 2 for arbitrary graphs and the case h ≥ 3 when the second factor of the corona product is an empty graph.As an example of a chemical compound whose graph is obtained as a corona product graph we consider the Cycloalkanes with a single ring, whose chemical formula is C k H 2k , and whose molecular graph can be expressed as C k N 2 , where C k is the cycle graph of order k and N 2 is the empty graph of order two.We recall that, given two graphs G and H of order n 1 and n 2 , respectively, the corona product G H is defined as the graph obtained from G and H by taking one copy of G and n 1 copies of H and then joining by an edge each vertex of the ith copy of H with the ith vertex of G.

Estimating R h for Corona Graphs
Theorem 2.1.For i ∈ {1, 2}, let G i be a graph of minimum degree δ i , maximum degree Δ i , order n i and size m i .Then, where

2.3
Thus, the lower bound follows.Analogously we deduce the upper bound.
Corollary 2.2.For i ∈ {1, 2}, let G i be a δ i -regular graph of order n i .Then, Theorem 2.3.For i ∈ {1, 2}, let G i be a graph of minimum degree δ i , maximum degree Δ i , order n i , and size m i .Then,

2.5
Proof.Let V 1 {v 1 , v 2 , . . ., v n 1 } and V 2 {u 1 , u 2 , . . ., u n 2 } be the set of vertices of G 1 and G 2 , respectively.Given a vertex v ∈ V i , we denote by N G i v the set of neighbors that v has in G i .
The paths of length two in G 1 G 2 are obtained as follows: iv paths of length two belonging to G 1 , v paths of length two belonging to the n 1 copies of G 2 .
So, we have ISRN Discrete Mathematics corresponds to the paths type i , corresponds to the paths type ii , corresponds to the paths type iii , 2.9 corresponds to the paths type iv , and corresponds to the paths type v .Thus, the lower bound follows.The upper bound is obtained by analogy.

2.11
The girth of a graph is the size of its smallest cycle.For instance, the molecular graphs of benzenoid hydrocarbons have girth 6.The molecular graphs of biphenylene and azulene have girth 4 and 5, respectively 11 .
The following result, and its proof, was implicitly obtained in the proof of Theorem 1 of 10 .By completeness, here we present it as a separate result.

Lemma 2.5. Let G
V, E be a graph with girth g G .If δ ≥ 2 and g G > h, then the number of paths of length h in G is bounded by

2.12
Proof.Since δ ≥ 2, for every v ∈ V , the number of paths of length 2 in G of the form v i vv j is δ v δ v − 1 /2.Therefore, the result follows for h 2. Suppose now that h ≥ 3. Given a vertex u ∈ V , let P h u be the set of paths of length h whose second vertex is u, that is, paths of the form u 1 uu 2 • • • u h .We denote by N v the set of neighbors of an arbitrary vertex v ∈ V .Note that the degree of v is δ v |N v |.If δ ≥ 2, then for every v ∈ V and w ∈ N v we have N w \ {v} / ∅.So, for every u ∈ V , there exists a vertex sequence Conversely, every path of length h whose second vertex is u can be constructed as above.Hence, the number of paths of length h whose second vertex is u is bounded by

2.13
Thus, the result follows.
Now N k denotes the empty graph of order k.
Theorem 2.6.Let G V, E be a graph with girth g G , minimum degree δ, and maximum degree

2.14
Proof.The paths of length

2.15
Moreover, each path of length h − 1 in G leads to 2k paths of length h in G N k ; thus, the paths of length h − 1 in

2.17
By Lemma 2.5 we obtain the upper bound and the lower bound is obtained by analogy.
Corollary 2.7.Let G V, E be a δ-regular graph of order n and girth g G .If δ ≥ 2 and g G > h ≥ 3, then