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

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
$-1\leq \alpha < -0.867$
. 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
$-1\leq \alpha < \alpha _{0}$
, 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.