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A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, we classify the s-regular elementary Abelian coverings of the three-dimensional hypercube for each s ≥ 1 whose fibre-preserving automorphism subgroups act arc-transitively. This gives a new infinite family of cubic 1-regular graphs, in which the smallest… (More)

We consider the following generalization of distance-regular digraphs. A connected digraph is said to be weakly distance-regular if, for all vertices x and y with (@(x; y); @(y; x)) = ˜ h, |{z ∈ VV | (@(x; z); @(z; x)) = ˜ i and (@(z; y); @(y; z)) = ˜ j}| depends only oñ h; ˜ i and˜j. We give some constructions of weakly distance-regular digraphs and… (More)

We introduce some constructions of weakly distance-regular digraphs of girth 2, and prove that a certain quotient digraph of a commutative weakly distance-transitive digraph of girth 2 is a distance-transitive graph. As an application of the result, we obtain some examples of weakly distance-regular digraphs which are not weakly distance-transitive.… (More)

Let G be a connected graph and H be an arbitrary graph. In this paper, we study the identifying codes of the lexicographic product G[H] of G and H. We first introduce two parameters of H, which are closely related to identifying codes of H. Then we provide the sufficient and necessary condition for G[H] to be identifiable. Finally, if G[H] is identifiable,… (More)

For any set S of positive integers, a mixed hypergraph H is a realization of S if its feasible set is S, furthermore, H is a one-realization of S if it is a realization of S and each entry of its chromatic spectrum is either 0 or 1. Jiang et al. showed that the minimum number of vertices of a realization of {s, t} with 2 ≤ s ≤ t − 2 is 2t − s. Král proved… (More)