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Antipodal Distance-transitive Covers of Complete Bipartite Graphs

✍ Scribed by A.A. Ivanov; Robert A. Liebler; Tim Penttila; Cheryl E. Praeger

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 480 KB
Volume: 18
Category: Article
ISSN: 0195-6698
DOI: 10.1006/eujc.1993.0086

No coin nor oath required. For personal study only.

✦ Synopsis

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 , where ( q b Ϫ 1) gcd ( b , q Ϫ 1) divides 2 a ( q Ϫ 1) and q is a prime power) .

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