Minimum second neighborhood degree energy of commuting graphs of finite rings

TL;DR

This paper calculates the minimum second neighborhood degree spectrum and energy of commuting graphs for specific finite non-commutative rings, proving related conjectures and classifying their spectral properties.

Contribution

It introduces new spectral results for commuting graphs of finite rings, confirming conjectures and analyzing their MSN-integrality and hyperintegrality.

Findings

Commuting graphs of certain finite rings are MSN-integral.

Proved conjectures related to neighborhood energy of commuting graphs.

Classified the spectral properties of these graphs.

Abstract

In this paper, we compute minimum second neighborhood degree spectrum and energy of commuting graphs of certain finite non-commutative rings. In particular, we consider non-commutative rings of order $p^2, p^3, p^4, p^5, p^2q$ and $p^3q$, where $p$ and $q$ are primes. We shall also show that the commuting graphs of these rings are MSN-integral but not MSN-hyperintegral. Finally, employing the techniques used in this paper, we prove Conjecture 3 of [Nath, R. K., Fasfous, W. N. T., Das, K. C. and Shang, Y. Common neighbourhood energy of commuting graphs of finite groups, {\em Symmetry} {\bf 13}(9), Article No. 1651, 2021.] and Conjecture 3.12 of [W. N. T. Fasfous and Nath, R. K. Common neighborhood spectrum and energy of commuting graphs of finite rings, \emph{ Palestine J. Math.} \textbf{13}(1), 66--76, 2024.]. We conclude this paper with two open problems.