A Characterization of the Graphs of Bilinear (d×d)-Forms over \(\mathbb{F}_2\)

Abstract

The bilinear forms graph denoted here by Bilq(d×e) is a graph defined on the set of (d×e)-matrices (ed) over \(\mathbb{F}_q\) with two matrices being adjacent if and only if the rank of their difference equals 1.

In 1999, K. Metsch showed that the bilinear forms graph Bilq(d×e), d≥3, is characterized by its intersection array if one of the following holds:

q=2 and ed+4

q≥3 and ed+3.

Thus, the following cases have been left unsettled:

q=2 and e∈{d,d+1,d+2,d+3}

q≥3 and e∈{d,d+1,d+2}.

In this work, we show that the graph of bilinear (d×d)-forms over the binary field, where d≥3, is characterized by its intersection array. In doing so, we also classify locally grid graphs whose μ-graphs are hexagons and their intersection numbers bi,ci are well-defined for all i=0,1,2.

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Correspondence to Alexander L. Gavrilyuk.

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Gavrilyuk, A.L., Koolen, J.H. A Characterization of the Graphs of Bilinear (d×d)-Forms over \(\mathbb{F}_2\). Combinatorica 39, 289–321 (2019). https://doi.org/10.1007/s00493-017-3573-4

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Mathematics Subject Classification (2010)

  • 05E30
  • 05B25
  • 51E20
  • 05C50