Antipodal Distance-transitive Covers of Complete Bipartite Graphs

Abstract

This paper completes the classification of antipodal distance-transitive covers of the complete bipartite graphs K k , k , where k у 3 . For such a cover the antipodal blocks must have size r р k .Although the case r ϭ k has already been considered , we give a unified treatment of r р k . We use deep group-theoretic results as well as representation-theoretic data about explicit linear groups and group coset geometries .Apart from the generic examples arising from finite projective spaces , there are three sporadic examples (arising from the outer automorphisms of the symmetric group S 6 and of the Mathieu group M 1 2 and one related to non-abelian Singer groups on PG 2 (4)) and an infinite family having solvable automorphism group (and with parameters r ϭ q b , k ϭ q a , whereand q is a prime power) . Smith [23] showed that there were only two types of partitions À which can be preserved by Aut G , for a finite imprimitive distance-transitive graph ⌫ of valency at least 3 , namely : (1) À ϭ ͕ ⌬ 1 , ⌬ 2 ͖ , and each edge of ⌫ joins a vertex of ⌬ 1 to a vertex of ⌬ 2 , in which case ⌫ is bipartite ; andHere we use the notationV , which is then called the antipodal partition of V . With any antipodal graph ⌫ we can associate a natural quotient graph , which we shall denote by ⌫ Ј , the vertex set of which is the antipodal partition of ⌫ , two antipodal blocks being adjacent in ⌫ Ј whenever they contain adjacent vertices of ⌫ . If ⌫ is antipodal and its antipodal blocks have size r , then ⌫ is called an r -fold antipodal co er of ⌫ Ј . Smith [23] showed , moreover , that if ⌫ is a finite antipodal distance-transitive graph then its antipodal quotient ⌫ Ј is also distance-transitive . Thus , as part of the problem of classifying finite distance-transitive graphs , all distance-transitive , antipodal covers of the known distance-transitive graphs must be determined . This has been done by J . van

On Completely Regular Codes

“…Note that, for i ∈ {0, 1, u} the codes C (i) and C (i) * are CT. Also, for m = 6, all codes C (i) and C (i) * are CT. In [43] for the graphs (to be distance transitive) coming from such codes (to be CT ) were obtained the following divisibility conditions: 2 m is a power of 2 i , or 2 m = 2 i , or 2 i − 1 divides 2m. Therefore, we conjecture that when one of such divisibility conditions is satisfied, then C (i) * is a CT code.…”

On the 486-Vertex Distance-Regular Graphs of Koolen-Riebeek and Soicher

“…Such a design has automorphism group with index (q n − 1)/(q − 1) in AΓL(n, q) (see [14]); this group is an index-2 subgroup of the automorphism group of the incidence graph. These graphs are in fact distance-transitive: see Ivanov et al [13] for an alternative construction from the projective space PG(n, q). However, the graphs are not determined by their parameters: for example, there are exactly four graphs with the parameters of that arising from AG(3, 3) [16], while more generally there are vast numbers of non-isomorphic graphs with these parameters (cf.…”

Smith’s Theorem and a characterization of the 6-cube as distance-transitive graph

“…If the antipodal classes had size greater than 3, then there would be cycles of odd length 3d in bipartite , which does not happen. This gives case (11). Now we additionally assume d = 2e Now assume d = 2e ≥ 8, and choose notation so that B A = AB .…”

“…The classification of distance-transitive antipodal covers of complete graphs, as in (2.9.6), and of complete bipartite graphs, as in (2.9.7), has been completely settled by, respectively, Godsil, Liebler, and Praeger in [7] and Ivanov, Liebler, Penttila, and Praeger in [11].…”