Laplacian Spectrum of cozero-divisor graphs of commutative polynomial rings

The cozero-divisor graph of a commutative ring $R$, denoted $\Gamma'(R)$, is the graph whose vertices are the non-zero and non-unit elements of $R$, with two distinct vertices $x$ and $y$ adjacent if and only if $x \notin Ry$ and $y \notin Rx$. This paper studies the structural properties of $\Gamma'(R)$ for the polynomial ring $R = \Z_n[x]/(x^2)$, where $n$ has the prime power decomposition of $p_1^{a_1}p_2^{a_2}\cdots p_q^{a_q}$. We provide a complete structure of the cozero-divisor graph for all $n$ 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 $n$. 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.