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

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