Antipodal Distance Transitive Covers of Complete Graphs

This paper is a contribution towards the determination of all finite distance-transitive graphs. We obtain a classification of all the antipodal distance-transitive graphs having as antipodal quotient a complete graph Kn . Such a graph necessarily has diameter 2 or 3 (see for example [2, Proposition 4.2.2 (ii)]). Those of diameter 2 are simply the complete multipartite graphs Kr,...,r with n parts of size r , and the heart of the classification lies in finding all the examples with diameter 3. In the diameter 3 case, the original graph 0 and the antipodal quotient have the same valency, and 0 is said to be a cover of its antipodal quotient. We offer the 3-fold cover of K5 that appears in Figure l as a motivating example. The antipodal quotient K5 is obtained by identifying vertices falling on lines through the centre of radial symmetry of the figure. More formally, this graph is the line graph of the Petersen graph or equivalently, the graph based on the involutions in the alternating group A5, two involutions being adjacent if their product has order 3. A 2-fold cover of Kn that is not bipartite is equivalent to a regular 2-graph, see [21] or [2, Theorem 1.5.3], and a result of Gardiner [8, Proposition 4.5] asserts that an (n − 1)-fold cover of Kn is equivalent to a Moore graph of valency n. Results of Gardiner [8], Taylor [22] and Aschbacher [1] together imply the classification of distance-transitive r -fold covers of Kn unless 3 ≤ r ≤ n − 2, and thus we need only deal with r in this range. The classification of the finite 2-transitive permutation groups is fundamental to our effort. Indeed our Lemma 2.6 shows that any such graph gives rise to two 2-transitive permutation groups and we play these two permutation groups off against each other to obtain our Main Theorem.