The graphs induced by maximal totally isotropic flats of affine-unitary spaces
Let \(\delta=0\) or \(1\), and let \(AUG(2\nu+\delta,\mathbb{F}_q)\) be the \((2\nu+\delta)\)-dimensional affine-unitary space over a finite field \(\mathbb{F}_q\). Let \(\Gamma\) be the graph whose vertex-set is the set of all maximal totally isotropic flats of \(AUG(2\nu+\delta,\mathbb{F}_q)\), and in which \(F_1\), \(F_2\) are adjacent if and only if \(\dim(F_1\cup F_2)=\nu+1\) for any \(F_1,F_2\in \Gamma\). Let \(V(\Gamma)\) be the vertex-set of \(\Gamma\) and let \(\partial(F_1,F_2)\) be the distance between vertices \(F_1\) and \(F_2\). For any \(F_1\in V(\Gamma)\), we use \(\Gamma_r(F_1)\) to denote the set of vertices \(F_2\) in \(\Gamma\) satisfying \(\partial(F_1,F_2)=r\). The authors show a number of properties of \(\Gamma\) such as: If \(1\leq r\leq\nu\), then \(|V(\Gamma)|=q^{\nu+\delta}\prod_{i=1}^\nu(q^{i+\delta-1/2}+1)\), and \(\Gamma\) is a vertex transitive graph. For any two vertices \(F_1,F_2\) of \(\Gamma\), \(\dim(F_1\cup F_2)=\nu+r\) if and only if \(\partial(F_1,F_2)=r\). In particular, \(\Gamma\) is a graph with diameter \(\nu+\delta\).
scientific article; zbMATH DE number 5541688