A note on eigenvalues of zero divisor graphs associated with commutative rings

TL;DR

This work corrects prior results on the zero-divisor graphs $\Gamma(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ by deriving the accurate adjacency spectrum, energy, and Zagreb indices. It establishes the graph’s structure via a partition into three vertex sets, shows that eigenvalues $-1$ and $0$ occur with multiplicities $p^{2}-3$ and $p^{3}-p^{2}-1$, and identifies the remaining three eigenvalues as roots of a cubic, leading to corrected energy bounds. The authors prove the graphs are non-hyperenergetic for all primes and hypoenergetic for $p\ge 3$, and they provide closed-form expressions for various topological indices, including a corrected $M_{2}$ value and several degree-based indices, while confirming the Hansen and Vuki\v{c}evi\c{c} conjecture for these graphs. The results refine the spectral and topological-characterization of these zero-divisor graphs and supply formulas enabling direct computation of indices in this family of rings.

Abstract

For a commutative ring $R,$ with non-zero zero divisors $Z^{\ast}(R)$. The zero divisor graph $Γ(R)$ is a simple graph with vertex set $Z^{\ast}(R)$, and two distinct vertices $x,y\in V(Γ(R))$ are adjacent if and only if $x\cdot y=0.$ In this note, we provide counter examples to the eigenvalues, the energy and the second Zagreb index related to zero divisor graphs of rings obtained in [Johnson and Sankar, J. Appl. Math. Comp. (2023), \cite{johnson}]. We correct the eigenvalues (energy) and the Zagreb index result for the zero divisor graphs of ring $\mathbb{Z}_{p}[x]/\langle x^{4} \rangle.$ We show that for any prime $p$, $Γ(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ is non-hyperenergetic and for prime $p\geq 3$, $Γ(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ is hypoenergetic. We give a formulae for the topological indices of $Γ(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ and show that its Zagreb indices satisfy Hansen and Vuki$\check{c}$cević conjecture \cite{hansen}.

Figures (2)

• Figure 1: Zero divisor graph of $\Gamma (\mathbb{Z}_{3}[x]/\langle x^{4} \rangle )$.
• Figure 2: Picture showing the increasing behaviour of $f(p)$ and $g(p)$.

Theorems & Definitions (17)

• Remark 2.1
• Theorem 2.2: Theorem 3.2, johnson
• Lemma 2.3: dasDM
• Lemma 2.4: bilalaims
• Theorem 2.5
• Remark 2.6
• Corollary 2.7
• Theorem 2.8
• Remark 2.9
• Proposition 2.10