Constructing 2-Arc-Transitive Covers of Hypercubes

TL;DR

This paper introduces symmetric bases in vector spaces with quadratic forms and uses them to construct 2-arc-transitive covers of hypercubes via Cayley graphs of extraspecial 2-groups, advancing understanding of symmetric graph coverings.

Contribution

It provides a new framework using symmetric bases to construct and analyze 2-arc-transitive covers of hypercubes through Cayley graphs of specific 2-groups.

Findings

Symmetric bases characterized by necessary and sufficient conditions.

Construction of normal Cayley graphs of extraspecial 2-groups.

Existence of 2-arc-transitive covers of hypercubes.

Abstract

We introduce the notion of a symmetric basis of a vector space equipped with a quadratic form, and provide a sufficient and necessary condition for the existence to such a basis. Symmetric bases are then used to study Cayley graphs of certain extraspecial 2-groups of order 2^{2r+1} (r\geq 1), which are further shown to be normal Cayley graphs and 2-arc-transitive covers of 2r-dimensional hypercubes.