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Automorphism groups of half cubes

Richard H. Hammack and Benjamin B. MacKinnon

Vol. 16 (2023), No. 2, 321–329

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 (n1)-dimensional simplices and (n1)-dimensional half cube polytopes.

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

half cube, graph automorphism group, hypercube, graph automorphism group
Mathematical Subject Classification
Primary: 05C25, 05C50, 05C99
Received: 28 February 2022
Revised: 4 June 2022
Accepted: 4 June 2022
Published: 26 May 2023

Communicated by Ronald Gould
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