On sharp bounds of the zero-order Randić index of certain unicyclic graphs

Let G be a simple connected graph and t be a given real number. The zero-order general Randić index α t ( G ) of G is defined as ∑ v ∈ V ( G ) d ( v ) t , where d ( v ) denotes the degree of v . In this paper, for any t , we characterize the graphs with the greatest and the smallest α t within two subclasses of connected unicyclic graphs on n vertices, namely, unicyclic graphs with k pendant vertices and unicyclic graphs with a k -cycle.