Distance-Transitive Graphs of Valency k, 8 ≤ k ≤ 13
Alexander A. Ivanov, Andrey V. Ivanov · Cambridge University Press eBooks · 1988
INTRODUCTION In Faradẑev et al. (1936) an algorithm for enumeration of intersection arrays of distance–transitive graphs (d.t.g.) having given valency k is described. This algorithm was used in Faradẑev et al. (1936) for classification of the d.t.g.'s of valency k = 5, 6 and 7. This paper is a logical continuation of Faradẑev et al. (1986). Here we supply the algorithm with some new feasibility conditions. The new version of the algorithm enable us to classify the d.t.g.'s of valency up to 13. All definitions and notations concerning d.t.g.'s can be found in Faradẑev et al. (1936) (see also Bannai & Ito (1984) and Brouwer et al. (1937)). Let us summarise briefly the history of classification of d.t.g.'s having small valences. Chronologically the first result in the area is the classification of cubic d.t.g.'s (k=3), obtained by Biggs & Smith (1971). Their scheme of classification can be generalised on the case of an arbitrary valency k ≥ 3 and in general consists of the following four steps: (1) to prove that the order of the vertex stabiliser in the automorphism group of d.t.g. having valency k is bounded by some value f = f(k); (2) to bound the diameter of a d.t.g. having valency k by some value D = D(k); (3) to enumerate all feasible intersection arrays for the given valency k; (4) to construct all graphs with intersection arrays found on the step (3).