The A-like matrices for a hypercube

TL;DR

This paper characterizes the structure of matrices related to hypercube graphs that commute with the adjacency matrix and have specific zero entries, providing bases and dimensions for their symmetric and antisymmetric parts.

Contribution

It introduces the concept of A-like matrices for hypercube graphs and explicitly decomposes their space into symmetric and antisymmetric components with known bases and dimensions.

Findings

Dimensions of symmetric and antisymmetric parts are D+1 and D choose 2.

Explicit bases for both parts are provided.

The structure of A-like matrices is fully characterized.

Abstract

Let $D$ denote a positive integer and let $Q_D$ denote the graph of the $D$-dimensional hypercube. Let $X$ denote the vertex set of $Q_D$ and let $A \in \MX$ denote the adjacency matrix of $Q_D$. A matrix $B \in \MX$ is called $A$-{\em like} whenever both (i) $BA = AB$; (ii) for all $x,y \in X$ that are not equal or adjacent, the $(x,y)$-entry of $B$ is zero. Let $\Al$ denote the subspace of $\MX$ consisting of the $A$-like elements. We decompose $\Al$ into the direct sum of its symmetric part and antisymmetric part. We give a basis for each part. The dimensions of the symmetric part and antisymmetric part are $D+1$ and ${D \choose 2}$, respectively.