Remark on general sum-connectivity index

Abstract

Let $G =(V,E)$, $V=\{1,2,\ldots ,n\}$, $E=\{e_1,e_2,\ldots,e_m\}$,
be a simple connected graph with $n$ vertices and $m$ edges with
vertex degree sequence $d_1\ge d_2 \ge \cdots \ge d_n>0$. If $i$th
and $j$th vertices are adjacent, it is denoted as $i\sim j$.
Topological degree-based index of graph $H_{\alpha} =\sum_{i\sim
j} (d_i+d_j)^{\alpha}$, where $\alpha$ is an arbitrary real
number, is referred to as general sum-connectivity index. In this
paper we prove inequality that connects invariants $H_{\alpha}$,
$H_{\alpha-1}$ and $H_{\alpha-2}$. Using that inequality, in some
special cases we obtain lower bounds for some other graph
invariants.