Laplacian Bounds for the Dissociation Number of Regular Graphs of Matrix Rings

Abstract

Let $Γ_n(q)$ be the graph whose vertices are the invertible matrices in $\Mat_n(\F_q)$, with two distinct matrices adjacent whenever their sum is singular. A dissociation set is a vertex set inducing a graph of maximum degree at most one. We study the dissociation number of $Γ_n(q)$ by embedding it as an induced subgraph of the total graph $T_n(q)$ on all of $\Mat_n(\F_q)$. A general Laplacian inequality for $k$-independent sets, together with an explicit character computation for the additive group of the matrix ring, gives parity-sensitive upper bounds. For fixed $n$, the resulting bound is of order at most $q^{n^2-n+1}$ for odd $q$ and at most $q^{n^2-2n+2}$ for even $q$. In particular, \[ \diss(Γ_n(q))\le q^{n^2-n+1}-1. \] In the other direction, the regular representation of the extension field $\F_{q^n}$ gives $\diss(Γ_n(q))\ge q^n-1$. We give complete proofs, including a self-contained derivation of the required matrix character sum, and determine the smallest case: $\diss(Γ_2(2))=3$.