Automorphism groups of half cubes

Hammack, Richard H.; MacKinnon, Benjamin B.

doi:10.2140/involve.2023.16.321

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Automorphism groups of half cubes

Richard H. Hammack and Benjamin B. MacKinnon

Vol. 16 (2023), No. 2, 321–329

DOI: 10.2140/involve.2023.16.321

Abstract

The $n$ -dimensional half cube is the graph whose vertices are the binary strings of length $n$ , with an even number of 1’s, and with two vertices adjacent if and only if they differ in exactly two positions. It is also the 1-skeleton of an $n$ -dimensional polytope (called the half cube polytope) whose cells are $(n - 1)$ -dimensional simplices and $(n - 1)$ -dimensional half cube polytopes.

We compute the automorphism groups of the half cube graphs by embedding their vertices in $ℝ^{n}$ and realizing the automorphism groups as subgroups of ${GL}_{n} (ℝ)$ . As a consequence, the automorphism group of a half cube graph coincides with its automorphism group as a polytope.

Keywords

half cube, graph automorphism group, hypercube, graph automorphism group

Mathematical Subject Classification

Primary: 05C25, 05C50, 05C99

Milestones

Received: 28 February 2022

Revised: 4 June 2022

Accepted: 4 June 2022

Published: 26 May 2023

Communicated by Ronald Gould

Authors

Richard H. Hammack
Department of Mathematics and Applied Mathematics Virginia Commonwealth University Richmond, VA United States

Benjamin B. MacKinnon
Department of Mathematics Brightpoint Community College Midlothian, VA United States

Vol. 16, No. 2, 2023