Laplacian Spectrum of cozero-divisor graphs of commutative polynomial rings

Sarbari Mitra; Soumya Bhoumik

Paper

Laplacian Spectrum of cozero-divisor graphs of commutative polynomial rings

Abstract

The cozero-divisor graph of a commutative ring

, denoted

, is the graph whose vertices are the non-zero and non-unit elements of

, with two distinct vertices

and

adjacent if and only if

and

. This paper studies the structural properties of

for the polynomial ring

, where

has the prime power decomposition of

. We provide a complete structure of the cozero-divisor graph for all

up to cubic prime power decompositions. Furthermore, we determine the Laplacian spectrum of these graphs. Finally, we discuss the connectivity of such a cozero-divisor graph of the polynomial rings for any

. Our work provides the first comprehensive spectral analysis of cozero-divisor graphs for non-local polynomial rings and establishes powerful new techniques for bridging commutative algebra with spectral graph theory.