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Let ? be a distance-regular graph with diameter d. For vertices x and y of ? at distance i, 1 i d, we deene the sets (x) \ ?(y). Then we say ? has the CAB j property, if the partition CAB i (x; y) = fC i (x; y); A i (x; y); B i (x; y)g of the local graph of y is equitable for each pair of vertices x and y of ? at distance i j. We show that if ? has the CAB… (More)

In this paper, triangle-free distance-regular graphs with diameter 3 and an eigenvalue θ with multiplicity equal to their valency are studied. Let Γ be such a graph. We first show that θ = −1 if and only if Γ is antipodal. Then we assume that the graph Γ is primitive. We show that it is formally self-dual (and hence Q-polynomial and 1-homogeneous), all its… (More)

Let Γ be a triangle-free distance-regular graph with diameter d ≥ 3, valency k ≥ 3 and intersection number a 2 = 0. Assume Γ has an eigenvalue with multiplicity k. We show that Γ 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 Γ is an antipodal cover of a strongly regular graph, which means that… (More)

We classify triangle-and pentagon-free distance-regular graphs with diameter d ≥ 2, valency k, and an eigenvalue multiplicity k. In particular, we prove that such a graph is isomorphic to a cycle, a k-cube, a complete bipartite graph minus a matching, folded k-cube, k odd and k ≥ 7. This is a generalization of the results of Nomura [10] and Yamazaki [13],… (More)

Antipodal covers of strongly regular graphs which are not necessarily distance-regular are studied. The structure of short cycles in an antipodal cover is considered. In most cases, this provides a tool to determine if a strongly regular graph has an antipodal cover. In these cases, covers cannot be distance-regular except when they cover a complete… (More)