Sharp bounds on the zeroth-order general Randić index of unicyclic graphs with given diameter

Let G be a simple connected graph and α be a given real number. The zeroth-order general Randić index 0 R α ( G ) is defined as ∑ v ∈ V ( G ) [ d G ( v ) ] α , where d G ( v ) denotes the degree of the vertex v of G . In this work, we give, for any α ( ≠ 0 , 1 ) , some sharp bounds on the zeroth-order general Randić index 0 R α of all unicyclic graphs with n vertices and diameter d .