On the graphs of a fixed cyclomatic number and order with minimum general sum-connectivity and Platt indices

Abeer M. Albalahi, Zhibin Du, Akbar Ali, Abdulaziz Mohammed Alanazi

Abstract: The general sum-connectivity and Platt indices of a graph G are defined by \({\text {SC}}_a(G)=\sum _{xy\in E(G)}(d_x+d_y)^a\) and \({\text {Pl}}_a(G)=\sum _{xy\in E(G)}(d_x+d_y-2)^a\), respectively, where a is a real number, E(G) indicates the edge set of G, and \(d_v\) indicates the degree of a vertex v of G. The cyclomatic number of G is the least number of edges required to be deleted from G to make it acyclic. If the maximum degree of G is less than 5, then G is referred to as a molecular graph. In this paper, the problem of determining graphs possessing the minimum values of the indices \({\text {SC}}_a\) and \({\text {Pl}}_a\) among all connected (molecular) graphs of order n and cyclomatic number t is solved for \(0<a<1\) and \(n\ge 2(t-1)\ge 2\) with \(n\ge 4\). It is proved that the difference between the maximum and minimum degrees of the aforementioned extremal graphs is at most 1.