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

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.