ON N-VERTEX CHEMICAL GRAPHS WITH A FIXED CYCLOMATIC NUMBER AND MINIMUM GENERAL RANDI´C INDEX

Abstract

"The general Randi´c index of a graph G is defined as Rα(G) = P uv∈E(G)(dudv)α, where du and dv denote the degrees of the vertices u and v, respectively, α is a real number, and E(G) is the edge set of G. The minimum number of edges of a graph G whose removal makes G as acyclic is known as the cyclomatic number and it is usually denoted by ν. A graph with the maximum degree at most 4 is known as a chemical graph. For ν = 0, 1, 2 and α > 1, the problem of finding graph(s) with the minimum general Randi´c index Rα among all n-vertex chemical graphs with the cyclomatic number ν has already been solved. In this paper, this problem is solved for the case when ν ≥ 3, n ≥ 5(ν − 1), and 1 < α < α0, where α0 ≈ 11.4496 is the unique positive root of the equation 4(8α − 6α) + 4α − 9α = 0."