Antipodal Distance Transitive Covers of Complete Graphs

AbstractA distance-transitive antipodal cover of a complete graphKnpossesses an automorphism group that acts 2-transitively on the fibres. The classification of finite simple groups implies a classification of finite 2-transitive permutation groups, and this allows us to determine all possibilities for such a graph. Several new infinite families of distance-transitive graphs are constructed

ON A CLASS OF EDGE-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS

The paper is devoted to the problem of classification of edge-transitive distance-regular antipodal covers of complete graphs. This extends the classification of those covers that are arc-transitive, which has been settled except for some tricky cases that remain to be considered, including the case of covers satisfying condition $c_2=1$ (which means that every two vertices at distance 2  have exactly one common neighbour).Here it is shown that an edge-transitive distance-regular antipodal cover of a complete graph with $c_2=1$ is either the second neighbourhood of a vertex in a Moore graph of valency 3 or 7, or a Mathon graph, or a half-transitive graph whose automorphism group induces an affine  $2$-homogeneous group on the set of its fibres. Moreover,  distance-regular  antipodal covers of complete graphs  with $c_2=1$ that admit  an automorphism group acting  $2$-homogeneously on the set of fibres (which turns out to be an approximation of the property of edge-transitivity  of such  cover), are described.   A well-known correspondence between distance-regular antipodal covers of complete graphs with $c_2=1$ and geodetic graphs of diameter two that can be viewed as underlying graphs of certain Moore geometries, allows us to effectively restrict admissible automorphism groups of covers under consideration by combining Kantor's classification of involutory automorphisms of these geometries together with the classification of finite 2-homogeneous permutation groups

ON SOME VERTEX-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS

In the present paper, we classify abelian antipodal distance-regular graphs $\Gamma$ of diameter 3 with the following property: $(*)$ $\Gamma$ has a transitive group of automorphisms $\widetilde{G}$ that induces a primitive almost simple permutation group $\widetilde{G}^{\Sigma}$ on the set ${\Sigma}$ of its antipodal classes. There are several infinite families of (arc-transitive) examples in the case when the permutation rank ${\rm rk}(\widetilde{G}^{\Sigma})$ of $\widetilde{G}^{\Sigma}$ equals 2 moreover, all such graphs are now known. Here we focus on the case ${\rm rk}(\widetilde{G}^{\Sigma})=3$.Under this condition the socle of $\widetilde{G}^{\Sigma}$ turns out to be either a sporadic simple group, or an alternating group, or a simple group of exceptional Lie type, or a classical simple group. Earlier, it was shown that the family of non-bipartite graphs $\Gamma$ with the property $(*)$ such that $rk(\widetilde{G}^{\Sigma})=3$ and the socle of $\widetilde{G}^{\Sigma}$ is a sporadic or an alternating group is finite and limited to a small number of potential examples. The present paper is aimed to study the case of classical simple socle for $\widetilde{G}^{\Sigma}$. We follow a classification scheme that is based on a reduction to minimal quotients of $\Gamma$ that inherit the property  $(*)$. For each given group $\widetilde{G}^{\Sigma}$ with simple classical socle of degree $|{\Sigma}|\le 2500$, we determine potential minimal quotients of $\Gamma$, applying some previously developed techniques for bounding their spectrum and parameters in combination with the classification of primitive rank 3 groups of the corresponding type and associated rank 3 graphs. This allows us to essentially restrict the sets of feasible parameters of $\Gamma$ in the case of classical socle for $\widetilde{G}^{\Sigma}$ under condition \(|{\Sigma}|\le 2500.\

On a Class of Edge-Transitive Distance-Regular Antipodal Covers of Complete Graphs

The paper is devoted to the problem of classification of edge-transitive distance-regular antipodal covers of complete graphs. This extends the classification of those covers that are arc-transitive, which has been settled except for some tricky cases that remain to be considered, including the case of covers satisfying condition c2=1 (which means that every two vertices at distance 2 have exactly one common neighbour). Here it is shown that an edge-transitive distance-regular antipodal cover of a complete graph with c2=1 is either the second neighbourhood of a vertex in a Moore graph of valency 3 or 7, or a Mathon graph, or a half-transitive graph whose automorphism group induces an affine 2-homogeneous group on the set of its fibres. Moreover, distance-regular antipodal covers of complete graphs with c2=1 that admit an automorphism group acting 2-homogeneously on the set of fibres (which turns out to be an approximation of the property of edge-transitivity of such cover), are described. A well-known correspondence between distance-regular antipodal covers of complete graphs with c2=1 and geodetic graphs of diameter two that can be viewed as underlying graphs of certain Moore geometries, allows us to effectively restrict admissible automorphism groups of covers under consideration by combining Kantor's classification of involutory automorphisms of these geometries together with the classification of finite 2-homogeneous permutation groups.This work was supported by the Russian Science Foundation under grant no. 20-71-00122

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page