Wiener Index of Total Graph of Some Graphs

Abstract

Let $G = (V, E)$ be a graph. The \textit{total graph} $T(G)$ of $G$ is that graph whose vertex set is $V \cup E$, and two vertices are adjacent if and only if they are adjacent or incident in $G$. For a graph $G=(V, E)$, the graph $G.S_m$ is obtained by identifying each vertex of $G$ by a root vertex of $S_m$ and the graph $S_m.G$ is obtained by identifying each vertex of $S_m$ except root vertex by any vertex of $G$, where $S_m$ is a star graph with $m$ vertices. In this paper, we consider $G$ as the cycle graph $C_n$ with $n$ vertices and investigate the Wiener index of the total graphs of $C_n.S_m$ and $S_n.C_m$.