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Combinatorica

, Volume 39, Issue 2, pp 289–321 | Cite as

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

  • Alexander L. GavrilyukEmail author
  • Jack H. Koolen
Article
  • 35 Downloads

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.

Mathematics Subject Classification (2010)

05E30 05B25 51E20 05C50 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPR China
  2. 2.N. N. Krasovskii Institute of Mathematics and MechanicsUral Branch of Russian Academy of SciencesYekaterinburgRussia
  3. 3.Wen-tsun Wu Key Laboratory of CAS School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPR China

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