Sharp bounds for the general Randić 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 $ \mathscr{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 $ \mathscr{G}_{n}^{\gamma} $ with minimum general Randić index for $ \gamma\geq 3 $ and $ 1\leq\alpha\leq \frac{39}{25} $. These extend the main result proved by A. Ali, K. C. Das and S. Akhter in 2022. The elements of $ \mathscr{G}_{n}^{\gamma} $ with maximum general Randić index were also completely determined for $ \gamma\geq 3 $ and $ \alpha\geq 1 $.