On the lower and upper bounds for general Randić index of chemical (n,m)-graphs

A chemical (n,m)-graph is a connected graph of order n, size m and maximum degree at most 4. The general Randić index of a graph is defined as the sum of the weights [d(u)d(v)]α of all edges uv of the graph, where α is any real number and d(u) is the degree of a vertex u. In this paper, we give the lower and upper bounds for general Randić index of chemical (n,m)-graphs. A graph of order n and size m is called an (n,m)-graph. A connected graph is called chemical if its maximum degree is at most 4. The Randić index of a graph G is defined in [6] as χ(G) = ∑ uv 1 [d(u)d(v)]1/2 , (1) ∗Corresponding author: [email protected] 1 where uv runs over all edges of G, and d(u) is the degree of a vertex u. The general Randić index is defined in [1, 2] as follows χα(G) = ∑ uv∈E(G) [d(u)d(v)], (2) where α is a real number. The authors of [3, 4] studied the lower and upper bounds for Randić index (i.e. α = −1/2) of chemical (n,m)-graphs, while the authors of [5] gave the bounds for chemical trees with α = −1. The focus of this paper is on the lower and upper bounds for general Randić index (i.e. any real number α) of chemical (n,m)-graphs. Suppose that G is a chemical (n,m)-graph. Let xij denote the number of edges each having end-vertices of degrees i and j respectively, for 1 ≤ i ≤ j ≤ 4. Note that G is connected, x11 is thus zero for n > 2. Therefore, (2) can be presented as χα(G) = ∑ 1≤i≤j≤4 (ij)xij = 2x12 + 3 x13 + 4 (x14 + x22) + 6 x23 + 8 x24 +9x33 + 12 x34 + 16 x44 . If we count the number of vertices and the number of edges in two different ways, respectively, we would then have the following identities,