Sharp bounds for the general Randic index of graphs with fixed number of vertices and cyclomatic number

The cyclomatic number, denoted by gamma, of a graph G is the minimum number of edges of G whose removal makes G acyclic. Let G(n)(gamma) be the class of all connected graphs with order n and cyclomatic number gamma. In this paper, we characterized the graphs in G(n)(gamma) with minimum general Randic index for gamma >= 3 and 1 <= alpha <= 39 /25. These extend the main result proved by A. Ali, K. C. Das and S. Akhter in 2022. The elements of G(n)(gamma) with maximum general Randic index were also completely determined for gamma >= 3 and alpha >= 1.