The Smith group of the hypercube

TL;DR

This paper computes the Smith group of the n-cube graph, providing insights into its algebraic structure by analyzing the elementary divisors of its adjacency matrix, which is valuable for understanding graph symmetries and properties.

Contribution

It introduces a method to determine the Smith group of the n-cube graph, a novel algebraic characterization of this well-studied combinatorial object.

Findings

The Smith group of the n-cube graph is explicitly computed.

The elementary divisors of the adjacency matrix are characterized.

Results enhance understanding of the algebraic structure of hypercube graphs.

Abstract

The $n$-cube graph is the graph on the vertex set of $n$-tuples of $0$s and $1$s, with two vertices joined by an edge if and only if the $n$-tuples differ in exactly one component. We compute the Smith group of this graph, or, equivalently, the elementary divisors of an adjacency matrix of the graph.