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

The general Randić index of a graph G is defined as R_α(G) = Σ_uv∈E(G)(d_ud_v)^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 Randić index R_α 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(8^α - 6^α) + 4^α - 9^α = 0.