On the adjacency spectrum of zero divisor graph of ring ℤn

摘要

The zero divisor graph Γ(R) of a commutative ring R with unity is a simple undirected graph whose vertices are all nonzero zero divisors of R and two distinct vertices x and y are adjacent if and only if xy=0. In this paper, we study the graphical structure and the adjacency spectrum of the zero divisor graph of ring ℤn. For any non-prime positive integer n≥4 with ξ number of proper divisors, we show that the adjacency spectrum of Γ(ℤn) consists of the eigenvalues of a symmetric matrix C(Υn) of size ξ×ξ, and at the most 0 and −1. Also, we find the exact multiplicity of the eigenvalue 0 and show that all eigenvalues of C(Υn) are nonzero, by determining the rank and nullity of the adjacency matrix of Γ(ℤn). We find the values of n for which the adjacency spectrum of Γ(ℤn) contains only nonzero eigenvalues. Finally, by computing the characteristic polynomial of the matrix C(Υn), we determine the characteristic polynomial of Γ(ℤn) whenever n is a prime power.