On the Minimal General Sum-Connectivity Index of Connected Graphs Without Pendant Vertices

Abstract

The general sum-connectivity index of a graph $G$ , denoted by $\chi _{_\alpha }(G)$ , is defined as $\sum _{uv\in E(G)}(d(u)+d(v))^{\alpha }$ , where $uv$ is the edge connecting the vertices $u,v\in V(G)$ , $d(w)$ denotes the degree of a vertex $w\in V(G)$ , and $\alpha $ is a non-zero real number. For $\alpha =-1/2$ and $n\geq 11$ , Wang et al. [On the sum-connectivity index, Filomat 25 (2011) 29–42] proved that $K_{2} + \overline {K}_{n-2}$ is the unique graph with minimum $\chi _{_\alpha }$ value among all the $n$ –vertex graphs having minimum degree at least 2, where $K_{2} + \overline {K}_{n-2}$ is the join of the 2-vertex complete graph $K_{2}$ and the edgeless graph $\overline {K}_{n-2}$ on $n-2$ vertices. Tomescu [2-connected graphs with minimum general sum-connectivity index, Discrete Appl. Math. 178 (2014) 135–141] proved that the result of Wang et al. holds also for $n\geq 3$ and $-1\leq \alpha . In this paper, it is shown that the aforementioned result of Wang et al. remains valid if the graphs under consideration are connected, $n\geq 6$ and $-1\leq \alpha , where $\alpha _{0}\approx -0.68119$ is the unique real root of the equation $\chi _{_\alpha }(K_{2} + \overline {K}_{4}) - \chi _{_\alpha }(C_{6})=0$ , and $C_{6}$ is the cycle on 6 vertices.