Some new bounds on the general sum--connectivity index

Document Type : Original paper


1 Knowledge Unit of Science University of Management and Technology, Sialkot 51310, Pakistan

2 Faculty of Electronic Engineering, 18000 Nis, Serbia

3 Faculty of Electronic Engineering, Nis, Serbia


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 $v_iin V$. With $isim j$ 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.


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