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The automorphism group and the convex subgraphs of the quadratic forms graph in characteristic 2. (English) Zbl 0789.05045

The quadratic forms graph \(\text{Quad} (n,q)\) has as vertices all quadratic forms on an \(n\)-dimensional vector space over \(\text{GF} (q)\). Two forms \(f\) and \(g\) are adjacent whenever \(\text{rank} (f-g)=1\) or 2. The authors determine the automorphism group of \(\text{Quad} (n,q)\) and describe all its convex (i.e. geodetically closed) subgraphs in case \(q\) is even. For odd \(q\) these problems have been solved respectively by L.-K. Hua [Ann. Math. 50, 8-31 (1949; Zbl 0034.157)] and E. W. Lambeck [Contributions to the theory of distance regular graphs, Ph. D. thesis, Technical University Eindhoven, 1990].

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05E30 Association schemes, strongly regular graphs

Citations:

Zbl 0034.157
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References:

[1] A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer, Berlin-Heidelberg, 1989. · Zbl 0747.05073
[2] Y. Egawa, “Association schemes of quadratic forms,” J. Combin. Theory Series A38 (1985), 1-14. · Zbl 0564.05014
[3] J. Hemmeter, and A. Woldar, “The complete list of maximal cliques of the Quad (<Emphasis Type=”Italic“>n, q), <Emphasis Type=”Italic“>q even,” preprint. · Zbl 0915.05111
[4] L.-K. Hua, “Geometry of symmetric matrices over any field with characteristic other than two,” Ann. Math.50 (1949), 8-31. · Zbl 0034.15701
[5] E.W. Lambeck, “Contributions to the theory of distance regular graphs,” Ph.D. thesis, Technical University Eindhoven, 1990. · Zbl 0758.05047
[6] M.-L. Liu, “Geometry of alternate matrices,” Acta Math. Sinica16 (1966), 104-135. · Zbl 0161.03301
[7] A. Munemasa, D.V. Pasechnik, and S.V. Shpectorov, “A local characterization of the graphs of alternating forms and the graphs of quadratic forms over <Emphasis Type=”Italic“>GF(2),” to appear.
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