Abstract
In this paper we present a general criteria to decide when the cycle \(C_n\) on \(n\) vertices and \(H_{n,1}\), the coalescence of the star \(S_{n-2}\) with the cycle \(C_3\), are extremal unicyclic graphs of a vertex-degree-based (VDB) topological index.
We show that many of the well known results on extremal values of VDB topological indices over unicyclic graphs can be obtained as particular cases of ours. Moreover, we obtain new results on extremal values of VDB topological indices,
such as the generalized Geometric-Arithmetic indices, the generalized Atom-Bond-Connectivity indices, and its exponentials, among others.
