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

• Main Authors: Shehnaz Akhter, Muhammad Imran
• Format: Article
• Language: English
• Published: SpringerOpen 2016-09-01
• Series: Journal of Inequalities and Applications
• Subjects: general sum-connectivity index; operation on graphs; cartesian product; total graph
• Online Access: http://link.springer.com/article/10.1186/s13660-016-1186-x

Abstract The general sum-connectivity index χ α ( G ) $\chi_{\alpha}(G)$ , for a (molecular) graph G, is defined as the sum of the weights ( d G ( a 1 ) + d G ( a 2 ) ) α $(d_{G}(a_{1})+d_{G}(a_{2}))^{\alpha}$ of all a 1 a 2 ∈ E ( G ) $a_{1}a_{2}\in E(G)$ , where d G ( a 1 ) $d_{G}(a_{1})$ (or d G ( a 2 ) $d_{G}(a_{2})$ ) denotes the degree of a vertex a 1 $a_{1}$ (or a 2 $a_{2}$ ) in the graph G; E ( 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 ) $S(G)$ , R ( G ) $R(G)$ , Q ( G ) $Q(G)$ , and T ( G ) $T(G)$ , and they also computed the Wiener index of these graph operations in terms of W ( F ( G ) ) $W(F(G))$ and W ( H ) $W(H)$ , 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.