Wiener index and addressing of the total graph

Abstract

Graham and Pollak showed that the vertices of any connected graph $G$ can be assigned $t$-tuples with entries in $\left\{0,a,b\right\}$, called addresses, such that the distance between any two vertices can be determined from their addresses. In this paper we determine the minimum value of such $t$, called squashed-cube dimension for the total graph $T\left(\mathrm{\Gamma }\left({\mathbb{Z}}_{2^n{p}^m}\right)\right)$ where $n,m\ge 1$ and $p\ge 3$ is a prime number.