Spectrum of the Unit-Graph on $\mathrm{Mat}_3(\mathbb{F}_q)$

TL;DR

This paper analyzes the spectrum of the unit-graph on 3x3 matrices over finite fields, revealing eigenvalues and demonstrating how large subsets necessarily contain invertible differences.

Contribution

It explicitly determines the adjacency spectrum of the unit-graph on Mat_3(F_q) and applies spectral gap results to combinatorial properties of large subsets.

Findings

The adjacency spectrum has four distinct eigenvalues with known multiplicities.

Large subsets of Mat_3(F_q) necessarily contain matrices with invertible differences.

Spectral gap results imply large sets cannot avoid having invertible differences.

Abstract

In this paper, we investigate the spectrum of the unit-graph of the ring of $3 \times 3$ matrices over a finite field $\mathbb{F}_q$, which is equivalently the Cayley digraph $ \mathrm{Cay}\!\left((\mathrm{Mat}_3(\mathbb{F}_q),+), \mathrm{GL}_3(\mathbb{F}_q)\right)$. This unit-graph has a vertex set $\mathrm{Mat}_3(\mathbb{F}_q)$ with a directed edge from $A$ to $B$ whenever $B - A \in \mathrm{GL}_3(\mathbb{F}_q)$. Then, two vertices are adjacent precisely when their difference is invertible. With relevant character theory, we consequently demonstrate that the adjacency spectrum of $ \mathrm{Cay}\!\left((\mathrm{Mat}_3(\mathbb{F}_q),+), \mathrm{GL}_3(\mathbb{F}_q)\right) $ consists of four distinct eigenvalues together with their multiplicities. Using the Spectral Gap Theorem for Cayley digraphs, we show that if two subsets of vertices in $\mathrm{Mat}_3(\mathbb{F}_q)$ are sufficiently large, then there are matrices in the two subsets whose difference lies in $\mathrm{GL}_3(\mathbb{F}_q)$. In particular, any sufficiently large subset of $\mathrm{Mat}_3(\mathbb{F}_q)$ contains two distinct matrices whose difference has nonzero determinant. This spectral gap implies that large vertex sets cannot avoid each other and must be connected by at least one edge.