one of the useful indices in molecular topology is randić index. the alternative double bonds andconjugation in the polyene compounds are one of the main properties in these compounds. someof the molecular orbital and structural properties refer to this specialty, such as: molar absorptioncoefficient (εmax) , electrical conductivity and difference energy level of the homo and lumoorbitals, etc....

Abstract In this paper, we give various lower and upper bounds for the energy of graphs in terms several topological indices graphs: first general multiplicative Zagreb index, Randić zeroth-order redefined indices, atom-bond connectivity index. Moreover, obtain new certain graph invariants as diameter, girth, algebraic radius.

Using the AutoGraphiX 2 system, Aouchiche, Hansen and Zheng [2] proposed a conjecture that the difference and the ratio of the Randić index and the diameter of a graph are minimum for paths. We prove this conjecture for chemical graphs.

The higher Randić index Rt (G) of a simple graph G is defined as Rt (G) = ∑ i1i2···it+1 1 √ δi1δi2 · · · δit+1 , where δi denotes the degree of the vertex i and i1i2 · · · it+1 runs over all paths of length t in G. In [J.A. Rodríguez, A spectral approach to the Randić index, Linear Algebra Appl. 400 (2005) 339–344], the lower and upper bound on R1(G) was determined in terms of a kind of Laplaci...

Let G be a simple graph with vertex set V (G) = {v1, v2, . . . , vn} and edge set E(G) = {e1, e2, . . . , em}. Similar to the Randić matrix, here we introduce the Randić incidence matrix of a graph G, denoted by IR(G), which is defined as the n × m matrix whose (i, j)-entry is (di) 1 2 if vi is incident to ej and 0 otherwise. Naturally, the Randić incidence energy IRE of G is the sum of the sin...

In this note, we derive the lower bound on the sum for Wiener index of bipartite graph and its bipartite complement, as well as the lower and upper bounds on this sum for the Randić index and Zagreb indices. We also discuss the quality of these bounds.

We establish some bounds for the number of spanning trees of connected graphs in terms of the number of vertices (n), the number of edges (m), maximum vertex degree (Δ1), minimum vertex degree (δ), first Zagreb index (M 1), and Randić index (R -1).