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

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Abstract

The bilinear forms graph denoted here by Bilq(d×e) is a graph defined on the set of (d×e)-matrices (e≥d) 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 e≥d+4

— q≥3 and e≥d+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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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, PR China

    Alexander L. Gavrilyuk

  2. N. N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, Kovalevskaya str., 16, Yekaterinburg, 620990, Russia

    Alexander L. Gavrilyuk

  3. Wen-tsun Wu Key Laboratory of CAS School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, PR China

    Jack H. Koolen

Authors
  1. Alexander L. Gavrilyuk
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  2. Jack H. Koolen
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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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  • DOI: https://doi.org/10.1007/s00493-017-3573-4

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