Properties of Total Transformation Graphs for General Sum-Connectivity Index

Abstract

The study of networks and graphs through structural properties is a massive area of research with developing significance. One of the methods used in studying structural properties is obtaining quantitative measures that encode structural data of the whole network by the real number. A large collection of numerical descriptors and associated graphs have been used to examine the whole structure of networks. In these analyses, degree-related topological indices have a significant position in theoretical chemistry and nanotechnology. Thus, the computation of degree-related indices is one of the successful topics of research. The general sum-connectivity $\left(\text{GSC}\right)$ index of graph $\mathcal{Q}$ is described as ${\chi }_{\rho }\left(\mathcal{Q}\right)={\displaystyle {\sum }_{\mathfrak{qq}\prime \in E\left(\mathcal{Q}\right)}{\left(d^{\left(\mathfrak{q}\right)}+d^{\left({\mathfrak{q}}^{\prime }\right)}\right)}^{\rho }}$ , where $d^{\left(\mathfrak{q}\right)}$ presents the degree of the vertex $\mathfrak{q}$ in $\mathcal{Q}$ and $\rho$ is a real number. The total graph $T\left(\mathcal{Q}\right)$ is a graph whose vertex set is $V\left(\mathcal{Q}\right)\cup E\left(\mathcal{Q}\right)$ , and two vertices are linked in $T\left(\mathcal{Q}\right)$ if and only if they are either adjacent or incident in $\mathcal{Q}$ . In this article, we study the general sum-connectivity index ${\chi }_{\rho }\left(\mathcal{Q}\right)$ of total graphs for different values of $\rho$ by using Jensen’s inequality.