Characteristic polynomial and Wiener index of the compressed zero divisor graph \(\Gamma_{E}[\mathbb {Z}_{p^n}]\)

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 relation on \(R\) given by \(r\backsim s\) if and only if ${\rm ann}_R(r) = {\rm ann}_R(s)\) is an equivalence relation. The compressed zero divisor graph \(\Gamma_E(R)\) is the (undirected) graph whose vertices are the equivalence classes induced by \(\backsim\) other than \([0]\) and \([1]\), such that distinct vertices \([r]\) and \([s]\) are adjacent in \(\Gamma_E(R)\) if and only if \(rs = 0\). We show that the coefficients of the compressed zero divisor graph \(\Gamma_{E}[\mathbb{Z}_{p^n}]\) form a Pascal like triangle and thus derive the characteristic polynomial in terms of binomial coefficients. We also calculate the Wiener index of \(\Gamma_{E}[\mathbb{Z}_{p^n}]\).