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Spectral Properties of Zero-Divisor Graphs of Truncated Polynomial Rings

Bilal Ahmad Rather

Abstract

Let $R$ be a commutative ring with identity and let $Z^{\ast}(R)$ denote the set of nonzero zero-divisors of $R$. The \emph{zero-divisor graph} $ \varGamma(R)$ is the simple graph with vertex set $V( \varGamma(R))=Z^{\ast}(R)$, where two distinct vertices$x,y\in Z^{\ast}(R)$ are adjacent if and only if $xy=0$ in $R$. In this paper we investigate the zero-divisor graph of the truncated polynomial ring $R=\mathbb{Z}_{p}[x]/\langle x^{c}\rangle,$ for $c\in\mathbb{N}.$ We determine the spectrum of the $A_α$-matrix associated with $ \varGamma(R)$, and, as special cases, explicitly obtain both the adjacency spectrum and the signless Laplacian spectrum of $ \varGamma(R)$. Furthermore, we prove that the Laplacian eigenvalues, as well as the distance eigenvalues, of these graphs are all integers.

Spectral Properties of Zero-Divisor Graphs of Truncated Polynomial Rings

Abstract

Let be a commutative ring with identity and let denote the set of nonzero zero-divisors of . The \emph{zero-divisor graph} is the simple graph with vertex set , where two distinct vertices are adjacent if and only if in . In this paper we investigate the zero-divisor graph of the truncated polynomial ring for We determine the spectrum of the -matrix associated with , and, as special cases, explicitly obtain both the adjacency spectrum and the signless Laplacian spectrum of . Furthermore, we prove that the Laplacian eigenvalues, as well as the distance eigenvalues, of these graphs are all integers.

Paper Structure

This paper contains 4 sections, 10 theorems, 180 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Spectra of zero-divisor graph of $\mathbb{Z}_p[x]/\langle x^{2b}\rangle$
  3. Spectra of zero-divisor graph of $\mathbb{Z}_p[x]/\langle x^{2b+1}\rangle$
  4. Comments

Key Result

Theorem 2.1

Let $p$ be a prime and let $a=2b$, where $b\in \mathbb{N}$. Let $R = \mathbb{Z}_p[x]/\langle x^{2b}\rangle\cong\mathbb{F}_p[x]/(x^{2b}),$ and $\varGamma(R)$ be its zero-divisor graph. Then the following hold. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: zero-divisor graph $\varGamma(R)$ for $R \cong \mathbb{F}_2[x]/(x^6)$.
  • Figure 2: Zero--divisor graph $\varGamma(R)$ for $R \cong \mathbb{Z}_2[x]/\langle x^5\rangle$.

Theorems & Definitions (12)

  • Theorem 2.1
  • Theorem 2.2
  • Example 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Theorem 3.1
  • Theorem 3.2
  • Example 3.3
  • Corollary 3.4
  • Corollary 3.5
  • ...and 2 more