Spectral properties of zero divisor graphs of commutative rings

Abstract

UDC 519.17; 512.55

Let \(R\) be a commutative ring with identity \(1 \neq 0 ,\) and let \(Z(R)\) be the set of zero divisors of \(R.\) The zero divisor graph \(\Gamma(R)\) is defined as a simple graph with the set of vertices formed by nonzero divisors \(Z^{+}(R) = Z(R) \setminus \{0\}\) of the zero element of $R$ such that two distinct vertices $ x $ and $ y $ are adjacent if and only if \(x \cdot y = 0.\) For a prime power $n,$ we obtain certain eigenvalues (eigenvectors), extremе bounds for the spectral spread, the largest and  smallest eigenvalues, and the number of distinct eigenvalues for $\Gamma(\mathbb{Z}_{n}).$ In addition, we complete the gaps in the spectral studies of zero divisor graphs detected in the existing literature by finding the determinant of the quotient matrices of zero divisor graphs and establish sharp bounds for the trace norm (energy) of zero divisor graphs of commutative and von Neumann regular rings.