Spectral Properties of Zero-Divisor Graphs of Truncated Polynomial Rings

Abstract: Let $R$ be a commutative ring with identity and let $Z^{\ast}(R)$ denote the set of nonzero zero-divisors of $R$. The \emph{zero-divisor graph} $ \varGamma(R)$ is the simple graph with vertex set $V( \varGamma(R))=Z^{\ast}(R)$, where two distinct vertices$x,y\in Z^{\ast}(R)$ are adjacent if and only if $xy=0$ in $R$. In this paper we investigate the zero-divisor graph of the truncated polynomial ring $R=\mathbb{Z}{p}[x]/\langle x^{c}\rangle,$ for $c\in\mathbb{N}.$ We determine the spectrum of the $Aα$-matrix associated with $ \varGamma(R)$, and, as special cases, explicitly obtain both the adjacency spectrum and the signless Laplacian spectrum of $ \varGamma(R)$. Furthermore, we prove that the Laplacian eigenvalues, as well as the distance eigenvalues, of these graphs are all integers.

• The paper computes the Aₐ-spectrum of zero-divisor graphs from truncated polynomial rings, bridging algebraic structure with graph theory.
• Methodologies include equitable partitioning and block diagonalization to derive explicit eigenvalue formulas for adjacency, Laplacian, and Aₐ matrices.
• Key results demonstrate that both Laplacian and distance Laplacian spectra are integral, supporting applications in spectral graph theory and combinatorial design.

Spectral Properties of Zero-Divisor Graphs of Truncated Polynomial Rings

Introduction and Context

This paper conducts a detailed spectral analysis of zero-divisor graphs arising from the truncated polynomial rings $R = \mathbb{Z}_p[x]/\langle x^c \rangle$, where $p$ is a prime and $c \in \mathbb{N}$. For a commutative ring $R$, the zero-divisor graph $\Gamma(R)$ is the simple graph with vertices corresponding to nonzero zero-divisors in $R$, with adjacency defined by the property $xy=0$ for $x\neq y$ in $R$. The motivation for examining such graphs stems from the interplay between algebraic ring structure and combinatorial and spectral properties of the associated graphs.

Beyond classical spectra (adjacency, Laplacian, signless Laplacian), the work addresses the more general $A_\alpha$-matrix spectrum—a one-parameter family $p$0, interpolating between important graph matrices. The main focus is to (1) compute the $p$1 spectrum of $p$2 for all $p$3, and (2) show that these graphs are Laplacian integral and distance Laplacian integral.

Structural Properties of Zero-Divisor Graphs

Let $p$4 with $p$5. The paper distinguishes the analysis based on the parity of $p$6:

• Even $p$7 ($p$8):

The vertex set of $p$9 is partitioned into $c \in \mathbb{N}$0 classes $c \in \mathbb{N}$1 according to the minimal degree of nonzero zero-divisors. The set $c \in \mathbb{N}$2 forms an independent set, while $c \in \mathbb{N}$3 forms a clique. The graph possesses $c \in \mathbb{N}$4 vertices, clique number $c \in \mathbb{N}$5 (the size of $c \in \mathbb{N}$6), an independence number $c \in \mathbb{N}$7, diameter $c \in \mathbb{N}$8, and girth $c \in \mathbb{N}$9 (for $R$0), with $R$1 containing universal vertices.

• Odd $R$2 ($R$3):

The vertex set is partitioned into $R$4 classes $R$5. For $R$6, $R$7 is an independent set; for $R$8, $R$9 is a clique. The maximal clique is of size $\Gamma(R)$0. The graph is always connected (via universal vertices in $\Gamma(R)$1), has diameter $\Gamma(R)$2, girth $\Gamma(R)$3 for $\Gamma(R)$4, clique number $\Gamma(R)$5, and independence number $\Gamma(R)$6.

$\Gamma(R)$7-Spectrum: Explicit Computation

Decomposition and Equitable Partition

For both even and odd $\Gamma(R)$8, an equitable partition of the vertex set according to the minimal degree allows for a block diagonalization approach to eigenvalue computation:

1. Eigenvalues from internal classes: For each $\Gamma(R)$9, the vectors supported entirely on $R$0 with coordinate sum zero are eigenvectors. Their eigenvalues can be given explicitly:
   • For independent set classes ($R$1 for $R$2, $R$3 for $R$4): eigenvalue $R$5.
   • For clique classes ($R$6 for $R$7, $R$8 for $R$9): eigenvalue $xy=0$0.
2. Remaining eigenvalues: The rest are determined by the restriction of $xy=0$1 to the $xy=0$2-dimensional space of vectors constant on each $xy=0$3 ($xy=0$4 or $xy=0$5), leading to the calculation of eigenvalues of an explicit quotient matrix $xy=0$6, parameterized by $xy=0$7, $xy=0$8, and $xy=0$9.

Strong claim: All eigenvalues corresponding to the internal classes and all blocks of $x\neq y$0 are explicitly determined in terms of $x\neq y$1, $x\neq y$2, and the sizes of the sets $x\neq y$3. The nature of the equitable partition ensures that the nontrivial spectra are always computable from the blocks.

Specializations

• Adjacency Spectrum: Setting $x\neq y$4, eigenvalues are $x\neq y$5 or $x\neq y$6, with remaining values as eigenvalues of the explicit quotient matrix $x\neq y$7.
• Signless Laplacian Spectrum: From $x\neq y$8, the spectrum is expressible in terms of values $x\neq y$9 ($R$0 in independent sets), $R$1 ($R$2 in cliques), and the spectrum of $R$3.
• Laplacian Spectrum: Using the identity connecting $R$4 for two values of $R$5, all Laplacian eigenvalues are integers for all $R$6 and $R$7.

Integrality and Distance Laplacian Spectra

Key result: Both the Laplacian spectrum and the distance Laplacian spectrum of these graphs are integral. Explicit formulas are provided, showing that the spectrum consists of $R$8 and values $R$9 (with suitable multiplicities), and, by a result of Aouchiche and Hansen, the distance Laplacian spectrum is determined as $A_\alpha$0 for $A_\alpha$1 the order of the graph.

Contradictory claim to existing intuition: Despite the non-regular, non-symmetric structure and nested equitable partitions, the Laplacian and distance Laplacian spectra are always completely integral and have explicit multiplicities based on the algebraic data of $A_\alpha$2.

Algebraic-Graph Theoretic Interplay and Implications

The analysis demonstrates a direct connection between the ideal structure in $A_\alpha$3 and the combinatorial and spectral structure of the corresponding zero-divisor graph. The partition by minimal degree encodes the influence of the nilpotency index and the structure of principal ideals in the truncated polynomial ring, translating directly into graph invariants (clique/independence number, diameter, etc.).

Practical implications:

The explicit knowledge of spectra allows for immediate calculation of spectral moments, graph energy, and application to chemistry-related indices for these graphs. The Laplacian and distance Laplacian integrality also enables the use of these graphs as examples and counterexamples in algebraic graph theory, particularly in the context of spectral characterizations and isospectral families.

Theoretical implications:

The paper extends the catalogue of infinite families of graphs for which integral Laplacian and distance Laplacian spectra can be confirmed, a nontrivial property with connections to combinatorial design, spectral graph theory, and algebraic combinatorics.

Prospects for Future Research

The explicit spectral formulas open the avenue for:

• Exploration of spectral characterizations for more general classes of quotient rings and their zero-divisor graphs;
• Investigation of dynamics of spectral invariants as $A_\alpha$4 and $A_\alpha$5 vary, emphasizing asymptotic properties and possible spectral convergence;
• Comparison with spectra of other commutative ring-based graphs (e.g., total graphs, comaximal graphs);
• Applications to extremal spectral graph theory within algebraic graph theory.

The block approach used here, facilitated by the algebraic structure, suggests potential generalizations to quotient rings with other types of ideals or more general local rings.

Conclusion

The paper gives a comprehensive and explicit spectral analysis of zero-divisor graphs for truncated polynomial rings over finite fields. It establishes the $A_\alpha$6-spectra, adjacency, Laplacian, signless Laplacian, and distance Laplacian spectra, establishing integrality in all these settings. These results illustrate the utility of algebraic techniques—particularly equitable partitions arising from ring structure—in producing spectral data and highlight a strongly regularizing effect of the ring's nilpotent hierarchy on graph spectra. The results have substantial implications for the broader algebraic graph theory of ring-based graphs and open the door for systematic spectral studies in related algebraic-combinatorial contexts.

Reference:

"Spectral Properties of Zero-Divisor Graphs of Truncated Polynomial Rings" (2604.03101)