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Almost 2-Homogeneous Graphs and Completely Regular Quadrangles

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Abstract

Many known distance-regular graphs have extra combinatorial regularities: One of them is t-homogeneity. A bipartite or almost bipartite distance-regular graph is 2-homogeneous if the number γ i  = |{x | ∂(u, x) = ∂(v, x) = 1 and ∂(w, x) = i − 1}| (i = 2, 3,..., d) depends only on i whenever ∂(u, v) = 2 and ∂(u, w) = ∂(v, w) = i. K. Nomura gave a complete classification of bipartite and almost bipartite 2-homogeneous distance-regular graphs. In this paper, we generalize Nomura’s results by classifying 2-homogeneous triangle-free distance-regular graphs. As an application, we show that if Γ is a distance-regular graph of diameter at least four such that all quadrangles are completely regular then Γ is isomorphic to a binary Hamming graph, the folded graph of a binary Hamming graph or the coset graph of the extended binary Golay code of valency 24. We also consider the case Γ is a parallelogram-free distance-regular graph.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, International Christian University, Mitaka, Tokyo, 181-8585, Japan

    Hiroshi Suzuki

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  1. Hiroshi Suzuki
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Correspondence to Hiroshi Suzuki.

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This research was partially supported by the Grant-in-Aid for Scientific Research (No.17540039), Japan Society of the Promotion of Science.

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Suzuki, H. Almost 2-Homogeneous Graphs and Completely Regular Quadrangles. Graphs and Combinatorics 24, 571–585 (2008). https://doi.org/10.1007/s00373-008-0812-x

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