Abstract
As a consequence of a famous theorem by Derek Smith, an unknown distance-transitive graph is either primitive of diameter at least two and valency at least three or is antipodal, bipartite, or both. In the imprimitive cases an unknown graph must have a primitive core of diameter at least two and valency at least three. It seems that the known list of primitive graphs is complete. Here, starting from earlier work by Hemmeter we find every bipartite distance-transitive double whose primitive halved is one of the known distance-transitive graphs of diameter two and valency at least three.
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Alfuraidan, M.R.: Imprimitive distance-transitive graphs, Ph.D. Thesis, Michigan State University (2004)
Alfuraidan, M.R.: Antipodal distance-transitive covers with primitive quotient of diameter two, manuscript (to appear soon)
Alfuraidan, M.R., Gardiner, A., Hall, J.I.: Untitled manuscript (in preparation)
Alfuraidan M.R., Hall J.I.: Imprimitive distance-transitive graphs with primitive core of diameter at least 3. Mich. Math J. 58, 31–77 (2009)
Alfuraidan M.R., Hall J.I.: Smith’s Theorem and a characterization of the 6-cube as distance-transitive graph. J. Algebraic Combin. 24, 195–207 (2006)
van Bon, J.T.M., Brouwer, A.E.: The distance-regular antipodal covers of classical distance-regular graphs. In: Colloq. Math. Soc. Janos Bolyai, Proc. Eger, 1988, pp. 141–166 (1987)
Brouwer A.E., Cohen A.M., Neumaier A.: Distance-Regular Graphs. Springer, Berlin (1989)
Godsil C., Royle G.: Algebraic Graph Theory. Springer, New York (2001)
Hemmeter J.: Halved graphs, Johnson and Hamming graphs. Utilitas Math. 25, 115–118 (1984)
Hemmeter J.: Distance-regular graphs and halved graphs. Eur. J. Combin. 7, 119–129 (1986)
Higman D.G.: Finite permutation groups of rank 3. Math. Z. 86, 145–156 (1964)
Smith D.H.: Primitive and imprimitive graphs. Quart. J. Math. Oxford 22(2), 551–557 (1971)
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Alfuraidan, M.R. Bipartite Distance-Transitive Doubles with Primitive Halved of Diameter Two. Graphs and Combinatorics 29, 1151–1174 (2013). https://doi.org/10.1007/s00373-012-1207-6
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DOI: https://doi.org/10.1007/s00373-012-1207-6