On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n

Bilal A Rather,Yilun Shang,Shariefuddin Pirzada,Tariq A Naikoo

doi:10.3390/math9050482

Bilal A Rather, Yilun Shang + Show 2 more

Open Access

https://doi.org/10.3390/math9050482

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Journal: Mathematics	Publication Date: Feb 26, 2021
Citations: 27	License type: CC BY 4.0

Affiliation: University of Kashmir, Northumbria University

Abstract

Given a commutative ring R with identity 1≠0, let the set Z(R) denote the set of zero-divisors and let Z*(R)=Z(R)∖{0} be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple graph whose vertex set is Z*(R) and each pair of vertices in Z*(R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ(Zn) for n=pN1qN2, where p<q are primes and N1,N2 are positive integers.

Highlights

All graphs considered in the present article are connected, undirected, simple and finite
The adjacency matrix A(G) = of G is a square matrix of order n, whose (i, j)-entry is 1, if vi and vj are adjacent and is 0, otherwise
· · · ≥ σn(M) are the singular values of M

Summary

Introduction

All graphs considered in the present article are connected, undirected, simple and finite. 2. Structure of the Zero-Divisor Graph Γ(ZpN1 qN2 ) We begin with the following definition. The compressed zero-divisor graph of a commutative ring R, denoted by ΓE(R), is the undirected, simple graph with the vertex set Z(RE) − {[0]}= RE − {[0], [1]} and is defined by RE = {[a] : a ∈ R}, where [a] = {b ∈ R : ann(a) = ann(b)} and the two vertices [a] and [b] are adjacent provided [a][b] = [0] = [ab].

Results

Conclusion