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

Abstract

For a commutative ring $R,$ with non-zero zero divisors $Z^{\ast }(R)$. The zero divisor graph $\Gamma (R)$ is a simple graph with vertex set $Z^{\ast }(R)$, and two distinct vertices $x,y\in V(\Gamma (R))$ 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 ${\mathbb{Z}}_p[x]/⟨x^4⟩.$ We show that for any prime $p$, $\Gamma ({\mathbb{Z}}_p[x]/⟨x^4⟩)$ is non-hyperenergetic and for prime $p\ge 3$, $\Gamma ({\mathbb{Z}}_p[x]/⟨x^4⟩)$ is hypoenergetic. We give a formulae for the topological indices of $\Gamma ({\mathbb{Z}}_p[x]/⟨x^4⟩)$ and show that its Zagreb indices satisfy Hansen and Vuki$\check{c}$cević conjecture \cite{hansen}.

20 pages, 3 Figures, Submitted to journal "Journal of Applied Mathematics and Computing" on 10 Apr 2023, Comments and suggestions are welcome and can be sent at [email protected]