Some new bounds on the general sum–connectivity index

Let $G=(V,E)$ be a simple connected graph with $n$ vertices, $m$ edges and sequence of vertex degrees $d_1 \ge d_2 \ge \cdots \ge d_n>0$, $d_i=d(v_i)$, where $v_i\in V$. With $i\sim j$ we denote adjacency of vertices $v_i$ and $v_j$. The general sum--connectivity index of graph is defined as $\chi_{\alpha}(G)=\sum_{i\sim j}(d_i+d_j)^{\alpha}$, where $\alpha$ is an arbitrary real number. In this paper we determine relations between $\chi_{\alpha+\beta}(G)$ and $\chi_{\alpha+\beta-1}(G)$, where $\alpha$ and $\beta$ are arbitrary real numbers, and obtain new bounds for $\chi_{\alpha}(G)$. Also, by the appropriate choice of parameters $\alpha$ and $\beta$, we obtain a number of old/new inequalities for different vertex--degree--based topological indices.

Abstract licence: CC BY-SA

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