Abstract
Terwilliger [15] has given the diameter bound d ≤ (s − 1)(k − 1) + 1 for distance-regular graphs with girth 2s and valency k. We show that the only distance-regular graphs with even girth which reach this bound are the hypercubes and the doubled Odd graphs. Also we improve this bound for bipartite distance-regular graphs. Weichsel [17] conjectures that the only distance-regular subgraphs of a hypercube are the even polygons, the hypercubes and the doubled Odd graphs and proves this in the case of girth 4. We show that the only distance-regular subgraphs of a hypercube with girth 6 are the doubled Odd graphs. If the girth is equal to 8, then its valency is at most 12.
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Koolen, J. On Subgraphs in Distance-Regular Graphs. Journal of Algebraic Combinatorics 1, 353–362 (1992). https://doi.org/10.1023/A:1022442717593
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DOI: https://doi.org/10.1023/A:1022442717593