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On edge but not vertex transitive regular graphs. (English) Zbl 0629.05040
Combinatorial design theory, Dedic. A. Rosa Occas. 50th Birthday, Ann. Discrete Math. 34, 273-285 (1987).
[For the entire collection see Zbl 0625.00011.]
A finite simple r-valent graph on v vertices is said to be semisymmetric if it is edge-transitive but not vertex-transitive. J. Folkman [J. Comb. Theory 3, 215-232 (1967; Zbl 0158.425)] first constructed semisymmetric graphs, gave numerous sufficient conditions and necessary conditions upon v for the existence of a semisymmetric graph with v vertices and posed many existence questions in terms of r and v, some of which have been solved. In the present paper the semisymmetric graphs with \(v\leq 28\) are enumerated by computer and nonexistence of a semisymmetric graph with \(v=30\) is shown by exhaustion (thus answering Folkman’s problem 4.2). The smallest unresolved existence question is now \(v=66\).
Reviewer: M.E.Watkins

MSC:
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)