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Full automorphism group of the generalized symplectic graph. (English) Zbl 1282.05214

Summary: Let \(\mathbb F_q\) be a finite field of odd characteristic, \(m,\nu\) the integers with \(1\leqslant m\leqslant\nu\) and \(K\) a \(2\nu\times 2\nu\) nonsingular alternate matrix over \(\mathbb F_q\). In this paper, the generalized symplectic graph \(GSp_{2\nu}(q,m)\) relative to \(K\) over \(\mathbb F_q\) is introduced. It is the graph with \(m\)-dimensional totally isotropic subspaces of the \(2\nu\)-dimensional symplectic space \(\mathbb F_q^{(2v)}\) as its vertices and two vertices \(P\) and \(Q\) are adjacent if and only if the rank of \(PKQ^T\) is 1 and the dimension of \(P\cap Q\) is \(m-1\). It is proved that the full automorphism group of the graph \(GSp_{2\nu}(q,m)\) is the projective semilinear symplectic group \(P\Sigma p(2\nu ,q)\).

MSC:

05E15 Combinatorial aspects of groups and algebras (MSC2010)
05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
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References:

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