On the adjacency spectrum of zero divisor graph of ring ℤn

Abstract

The zero divisor graph [Formula: see text] of a commutative ring [Formula: see text] with unity is a simple undirected graph whose vertices are all nonzero zero divisors of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study the graphical structure and the adjacency spectrum of the zero divisor graph of ring [Formula: see text]. For any non-prime positive integer [Formula: see text] with [Formula: see text] number of proper divisors, we show that the adjacency spectrum of [Formula: see text] consists of the eigenvalues of a symmetric matrix [Formula: see text] of size [Formula: see text], and at the most [Formula: see text] and [Formula: see text]. Also, we find the exact multiplicity of the eigenvalue [Formula: see text] and show that all eigenvalues of [Formula: see text] are nonzero, by determining the rank and nullity of the adjacency matrix of [Formula: see text]. We find the values of [Formula: see text] for which the adjacency spectrum of [Formula: see text] contains only nonzero eigenvalues. Finally, by computing the characteristic polynomial of the matrix [Formula: see text], we determine the characteristic polynomial of [Formula: see text] whenever [Formula: see text] is a prime power.