On triangle-free distance-regular graphs with an eigenvalue multiplicity equal to the valency

Abstract

Let Gamma be a triangle-free distance-regular graph with diameter d >= 3, valency k >= 3 and intersection number a(2) not equal 0. Assume Gamma has an eigenvalue with multiplicity k. We show that Gamma is 1-homogeneous in the sense of Nomura when d = 3 or when d >= 4 and a(4) = 0. In the latter case we prove that r is an antipodal cover of a strongly regular graph, which means that it has diameter 4 or 5. For d = 5 the following infinite family of feasible intersection arrays: {2 mu(2) + mu, 2 mu(2) + mu -1, mu(2), mu,1; 1, mu, mu(2), 2 mu(2) + mu - 1, 2 mu(2) + mu}, mu is an element of N, is known. For mu = 1 the intersection array is uniquely realized by the dodecahedron. For mu = 1 we show that there are no distance-regular graphs with this intersection array. (c) 2007 Elsevier Ltd. All rights reserved.