The characteristic equation and Wiener index of a compressed zero divisor graph

Abstract

The Zero divisor Graph of a commutative ring $R$, denoted by $\Gamma [R]$, is a graph whose vertices are non-zero zero divisors of R and two vertices are adjacent if their product is zero. The compressed zero divisor graph ${\Gamma }_E[R]$ is the (undirected) graph whose vertices are the equivalence classes such that distinct vertices [r] and [s] are adjacent if and only if rs = 0. In this paper we derive the characteristic polynomial and Wiener index of the Compressed zero divisor graph ${\Gamma }_E[{\mathbb{Z}}_m]$ where $m=p^n$ with prime $p$.