Some Notes on Randi ć Index

Some bounds are reported for Randić index in [10]. These bounds are obtained in a triangle-free graph. In [7], Randić index is investigated for molecular graphs and it is calculated for some graph classes in [8]. Considering the fact that these studies can be extended to the weighted graphs, new inequalities for weighted Randić index of a graph are obtained in this work. In [1], additively weighted Harary index of some composite graphs is formulated. We establish the weighted Randić index of some composite graphs in this study. Also, some special bounds are found for the energy of graphs in [3] and [4]. Using these results, some definitions and results for Randić index of weighted graphs are obtained. In addition, general Randić index is defined and some bounds for this index are given.


Introduction
Randić index is one of the most important tools of graph theory with applications in Mathematics, Physics and Chemistry. This graph theoretical index is invented by Milan Randic in [10]. He defined a molecular structure descriptor and composed a new summation using degrees of vertices. By d i , we denote the degree of the vertex v i in G. Randić index of a graph G is defined by where summation goes over all pairs of adjacent vertices of the underlying (molecular) graph.
Some bounds are reported for Randić index in [10]. These bounds are obtained in a triangle-free graph. In [7], Randić index is investigated for molecular graphs and it is calculated for some graph classes in [8]. Considering the fact that these studies can be extended to the weighted graphs, new inequalities for weighted Randić index of a graph are obtained in this work. In [1], additively weighted Harary index of some composite graphs is formulated. We establish the weighted Randić index of some composite graphs in this study. Also, some special bounds are found for the energy of graphs in [3] and [4]. Using these results, some definitions and results for Randić index of weighted graphs are obtained. In addition, general Randić index is defined and some bounds for this index are given.

Preliminaries
Let G be a simple connected graph with vertex set V (G) and edge set E(G). The symmetric square matrix R = R(G) of order n defined by ; otherwise  is called the Randić matrix of the graph G. Graph operations enable us to calculate some property of a large graph in terms of some smaller graphs. Some fundamental information about the graph operations and their properties can be found in [6], [2], [11] and [12]. In the third section, we introduce new inequalities about the weighted Randić index of some graph operations. We also generalize Randić index and we give some bounds related to this new generalized index. We define the weighted Randić matrix and weighted Randić energy and obtain an upper bound for the second greatest eigenvalue ρ w 2 of the weighted Randić matrix. The Randić energy of a graph G is defined by where ρ 1 , ρ 2 , · · · , ρ n are the eigenvalues of its Randić matrix. We recall the following result:

Randić Index of Graph Operations
A weighted Randić index is obtained by adding a weight to each edge that is defined in [6] as follows: Let G be a simple, connected and weighted graph having n vertices. Let each edge of G be weighted with positive real numbers. The weighted Randić index R w = R w (G) of G is defined as follows: where w(u) is the sum of the weights on u and w(v) is the sum of the weights on v, that is, w(u) = i∼u t i , t i are the weights for i = 1, 2, · · · , n and w(v) = j∼v c j , c j are the weights for j = 1, 2, · · · , n.
We now recall some graph operations we shall need in this paper. Such operations help us to Let G and H be two simple graphs. The sum G+H of these two graphs is defined as the graph having the vertex set V (G+H) = V (G)∪V (H) and the edge set

The composition of two graphs G and H is denoted by G[H] and it is the graph with vertex set
and u 2 and v 2 are adjacent in H). It is obtained by connecting every vertex of G into a copy of H and replace every edge of G by all possible edges between the copies of H that arose from its end-vertices.
The symmetric difference G ⊕ H of G and H is the graph having vertex set V (G) × V (H) and edge set We can now calculate some indices: Theorem 3.1. Let G and H be two simple weighted graphs. Then the weighted Randić index of the sum of G and H is We partition the set of pairs of vertices of G + H to obtain the following three sums denoted by S 1 , S 2 , S 3 , respectively. Firstly, for each sum, we consider w i as the sum of the weights in each vertex i. In S 1 , we collect all pairs of vertices i and j so that i, j are in V (G) and ij is in E(G). Hence, i and j are adjacent vertices in E(G). For S 1 , we obtain, .
For the second sum S 2 , we take the vertices i and j in V (H) so that ij is in E(H). Hence, .
In the third sum S 3 , i is taken in V (G) and j is in V (H). So, .
The result now follows by adding the three contributions and simplifying the resulting expression.
Similarly, the weighted Randić indices of some other operations are obtained as follows: Theorem 3.2. Let G and H be two simple weighted graphs with edge sets E(G) and E(H) and vertex sets V (G) and V (H), respectively. The weighted Randić indices of the composition, symmetric difference, cartesian product and union of graphs G and H are respectively given by Proof. We denote by w(i) the sum of the weights in each vertex i. For each vertex i of G, we name the corresponding copy of H by H i . If two vertices i, j of G are adjacent, then every pair of vertices of H i and H j are adjacent, too. Hence, .
We partition this sum into three sums S 1 , S 2 and S 3 as follows: The first one S 1 runs over all pairs of e ij in E(G) ∩ E(H) for each vertex pair i, j in V (G) ∩ V (H). Hence, The second one S 2 is over all pairs ik such that ik is not in The third one S 3 is over all pairs ij such that ij is not in V (G)∩V (H) for each vertex i, j in V (G)∪V (H). Hence, Now the result follows. The third and fourth indices follow by the definitions of the operations.

On The General Randić Index
In this section, we obtain inequalities giving upper and lower bounds for general Randić index of a graph G which is defined by for α ∈ R. We first have Theorem 3.3. Let G be a nontrivial connected graph and let a, b ∈ R. Then, which proves the second inequality. For the first expression, we have four cases: Hence (R a+b (G)) 1 2 (R a−b (G)) 1 2 ≤ k a,b (R a (G)), R a (G) ≥ k a,b R a+b (G)R a−b (G).
If (a + b)(a − b) < 0, then, Hence, R a (G) ≥ k a,b R a+b (G)R a−b (G).

The Eigenvalues of Randić Matrix
In this section we will define and study the weighted Randić matrix and correspondingly, the weighted Randić energy of a graph. Also, we will establish a bound in terms of the weighted Randić energy and the eigenvalues of the weighted Randić matrix.
Definition 3.1. Let G be a simple, connected and weighted graph. The weighted Randić matrix W R = W R(G) of G is defined by ; otherwise and the weighted Randić eigenvalues ρ w 1 , ρ w 2 , · · · , ρ w n are defined as the eigenvalues of the Randić matrix W R.