On vertex-transitive distance-regular covers of complete graphs with an extremal smallest eigenvalue

TL;DR

This paper classifies certain vertex-transitive abelian covers of complete graphs that produce extremal eigenvalues and equiangular line sets, expanding understanding of their symmetry and spectral properties.

Contribution

It provides a classification of abelian distance-regular covers with vertex-transitive automorphism groups and extremal smallest eigenvalues, using permutation group theory.

Findings

Classified covers with automorphism groups of rank at most 3.

Identified conditions for extremal eigenvalues in these covers.

Connected the covers to equiangular line sets in complex Hilbert spaces.

Abstract

The paper is devoted to the study of abelian (in the sense defined by Godsil and Hensel) distance-regular $r$-covers of the complete graphs $K_n$. According to the construction by Coutinho, Godsil, Shirazi, and Zhan (2016), each such cover yields an equiangular set of lines of size $n$ that attains the relative bound. Moreover, there are four families of abelian covers that, through this construction, yield sets of lines attaining the absolute bound. All known representatives of these families -- the hexagon, the icosahedron graph, Taylor extensions of the Schl\"afli and McLaughlin graphs together with their distance-$2$ graphs, and three other examples arising from generalized quadrangles -- have a vertex-transitive automorphism group with at most two orbits on the arc set of the cover. We aim to classify the covers from these families under the condition that the automorphism group of the cover is vertex-transitive and has at most two orbits on its arc set, which holds precisely when this group induces a transitive permutation group of rank at most $3$ on the set of cover fibres. We apply several fundamental classification results on permutation groups of rank at most $3$ to describe the family of covers for which $r$ is odd and the smallest eigenvalue is extremal, equaling $-\sqrt{\sqrt{n}+1}$, which corresponds to lines in a complex Hilbert space. The results obtained encompass cases where the automorphism group of the cover induces a primitive or imprimitive group of rank at most $3$ on the set of fibres.