On Normalized Laplacian Spectra of the Weakly Zero-Divisor Graph of the Ring ℤ_n

Abstract

For a finite commutative ring R with identity 1≠0, the weakly zero-divisor graph of R denoted as WΓ(R) is a simple undirected graph having vertex set as a set of non-zero zero-divisors of R and two distinct vertices a and b are adjacent if and only if there exist elements r∈ann(a) and s∈ann(b) satisfying the condition rs=0. The zero-divisor graph of a ring is a spanning sub-graph of the weakly zero-divisor graph. This article finds the normalized Laplacian spectra of the weakly zero-divisor graph WΓ(R). Specifically, the investigation is carried out on the weakly zero-divisor graph WΓ(Zn) for various values of n.

Keywords

• normalized Laplacian spectra
• weakly zero-divisor graph
• ring of integers modulo n
• Euler totient function