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

The general sum-connectivity index of a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, denoted by <inline-formula> <tex-math notation="LaTeX">$\chi _{_\alpha }(G)$ </tex-math></inline-formula>, is defined as <inline-formula> <tex-math notation="LaTeX">$\sum _{uv\in E(G)}(d(u)+d(v))^{\alpha }$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$uv$ </tex-math></inline-formula> is the edge connecting the vertices <inline-formula> <tex-math notation="LaTeX">$u,v\in V(G)$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$d(w)$ </tex-math></inline-formula> denotes the degree of a vertex <inline-formula> <tex-math notation="LaTeX">$w\in V(G)$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> is a non-zero real number. For <inline-formula> <tex-math notation="LaTeX">$\alpha =-1/2$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$n\geq 11$ </tex-math></inline-formula>, Wang <italic>et al.</italic> [On the sum-connectivity index, Filomat 25 (2011) 29–42] proved that <inline-formula> <tex-math notation="LaTeX">$K_{2} + \overline {K}_{n-2}$ </tex-math></inline-formula> is the unique graph with minimum <inline-formula> <tex-math notation="LaTeX">$\chi _{_\alpha }$ </tex-math></inline-formula> value among all the <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>–vertex graphs having minimum degree at least 2, where <inline-formula> <tex-math notation="LaTeX">$K_{2} + \overline {K}_{n-2}$ </tex-math></inline-formula> is the join of the 2-vertex complete graph <inline-formula> <tex-math notation="LaTeX">$K_{2}$ </tex-math></inline-formula> and the edgeless graph <inline-formula> <tex-math notation="LaTeX">$\overline {K}_{n-2}$ </tex-math></inline-formula> on <inline-formula> <tex-math notation="LaTeX">$n-2$ </tex-math></inline-formula> vertices. Tomescu [2-connected graphs with minimum general sum-connectivity index, Discrete Appl. Math. 178 (2014) 135–141] proved that the result of Wang <italic>et al.</italic> holds also for <inline-formula> <tex-math notation="LaTeX">$n\geq 3$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$-1\leq \alpha < -0.867$ </tex-math></inline-formula>. In this paper, it is shown that the aforementioned result of Wang <italic>et al.</italic> remains valid if the graphs under consideration are connected, <inline-formula> <tex-math notation="LaTeX">$n\geq 6$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$-1\leq \alpha < \alpha _{0}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\alpha _{0}\approx -0.68119$ </tex-math></inline-formula> is the unique real root of the equation <inline-formula> <tex-math notation="LaTeX">$\chi _{_\alpha }(K_{2} + \overline {K}_{4}) - \chi _{_\alpha }(C_{6})=0$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$C_{6}$ </tex-math></inline-formula> is the cycle on 6 vertices.