On zero divisor graphs of the rings $$Z_n$$

Abstract

For a commutative ring R with non-zero zero-divisor set $$Z^*(R)$$, the zero-divisor graph of R is $$\varGamma (R)$$ with vertex set $$Z^*(R)$$, where two distinct vertices x and y are adjacent if and only if $$xy=0$$. The zero-divisor graph structure of $${\mathbb {Z}}_{p^n}$$ is described. We determine the clique number, degree of the vertices, size, metric dimension, upper dimension, automorphism group, Wiener index of the associated zero-divisor graph of $${\mathbb {Z}}_{p^n}$$. Further, we provide a partition of the vertex set of $$\varGamma ({\mathbb {Z}}_{p^n})$$ into distance similar equivalence classes and we show that in this graph the upper dimension equals the metric dimension. Also, we discuss similar properties of the compressed zero-divisor graph.