On zeroth-order general Randić index of quasi-tree graphs containing cycles

Let GG be a simple graph with the vertex set V(G)V(G) and αα be a real number with α≠0α≠0. The zeroth-order general Randić index of GG is defined as Rα0(G)=∑v∈V(G)dα(v), where d(v)d(v) denotes the degree of the vertex vv in GG. A graph GG is called a quasi-tree graph, if there exists a vertex u∈V(G)u∈V(G) such that G[V(G)∖{u}]G[V(G)∖{u}] is a tree. In this paper, we characterize the extremal quasi-tree graphs containing cycles with the minimum and maximum values of the zeroth-order general Randić index for αα in different intervals.