The sharp bounds on general sum-connectivity index of four operations on graphs

Abstract

The general sum-connectivity index $\chi_{\alpha}(G)$ , for a (molecular) graph G, is defined as the sum of the weights $(d_{G}(a_{1})+d_{G}(a_{2}))^{\alpha}$ of all $a_{1}a_{2}\in E(G)$ , where $d_{G}(a_{1})$ (or $d_{G}(a_{2})$ ) denotes the degree of a vertex $a_{1}$ (or $a_{2}$ ) in the graph G; $E(G)$ denotes the set of edges of G, and α is an arbitrary real number. Eliasi and Taeri (Discrete Appl. Math. 157:794-803, 2009) introduced four new operations based on the graphs $S(G)$ , $R(G)$ , $Q(G)$ , and $T(G)$ , and they also computed the Wiener index of these graph operations in terms of $W(F(G))$ and $W(H)$ , where F is one of the symbols S, R, Q, T. The aim of this paper is to obtain sharp bounds on the general sum-connectivity index of the four operations on graphs.