Abstract
Enomoto-Mena[1] showed that two one-parameter families of distance-regular digraphs of girth 4 could possibly exist. Subsequently Liebler-Mena[2] found an infinite family of such digraphs generated over an extension ring ofZ/4Z. We prove that there are no other solutions except for multiplication by principal units to generate distance-regular digraphs of girth 4 under their method. In order to prove this, we introduce Gauss sums and three kinds of Jacobi sums over an extension ring ofZ/4Z. We give necessary and sufficient conditions for the existence of these digraphs under that method. It turns out that the Liebler-Mena solutions are the only solutions which satisfy the necessary and sufficient conditions. This fact has been conjectured for a time, but has never been proved.
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References
Enomoto, H., Mena, R.A.: Distance-regular digraphs of girth 4. J. Comb. Theory, Ser. B43, 293–302 (1987)
Liebler, R.A., Mena, R.A.: Certain distance-regular digraphs and related rings of characteristic 4. J. Comb. Theory, Ser. A47, 111–123 (1988)
Mann, H.B.: Addition Theorems. New York: Krieger 1976
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Yamada, M. Distance-regular digraphs of girth 4 over an extension ring of Z/4Z. Graphs and Combinatorics 6, 381–394 (1990). https://doi.org/10.1007/BF01787706
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DOI: https://doi.org/10.1007/BF01787706