A characterization of the graphs of bilinear (d× d)-forms over \mathbbF₂

2015/11/30 by Gavrilyuk, Alexander L., Koolen, Jack H.
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1511.09435

Abstract

The bilinear forms graph denoted here by Bil_q(e× d) is a graph defined on the set of (e× d)-matrices (e≥ d) over \mathbbF_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 Bil_q(e× d) 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 the intersection numbers b_i,c_i are well-defined for all i=0,1,2.

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