For the Hermitian variety $\mathcal{H}_r=H(r,q^2)$ of ${\rm{PG}}(r,q^2)$ we show that the associated linear representation graph $\Gamma_r^*({\mathcal{H}})$ is isomorphic to an affine polar graph and we use this isomorphism to determine the full automorphism group of $\Gamma_r^*({\mathcal{H}})$. We provide a classification of maximal cliques of $\Gamma_r^*({\mathcal{H}})$ in geometric terms showing that their length is equal to $q^{r}$ for $r$ even and $q^{r+1}$ for $r$ odd. We also show that maximal cliques of $\Gamma_r^*({\mathcal{H}})$ are not equivalent under the action of ${\rm Aut}(\Gamma_r^*({\mathcal{H}}))$ and they fall into $\frac{1}{2}r+1$ and $\frac{1}{2}(r+3)$ classes according as $r$ is even or odd.

Strongly regular graphs arising from Hermitian varieties

MARINO, Giuseppe;
2015

Abstract

For the Hermitian variety $\mathcal{H}_r=H(r,q^2)$ of ${\rm{PG}}(r,q^2)$ we show that the associated linear representation graph $\Gamma_r^*({\mathcal{H}})$ is isomorphic to an affine polar graph and we use this isomorphism to determine the full automorphism group of $\Gamma_r^*({\mathcal{H}})$. We provide a classification of maximal cliques of $\Gamma_r^*({\mathcal{H}})$ in geometric terms showing that their length is equal to $q^{r}$ for $r$ even and $q^{r+1}$ for $r$ odd. We also show that maximal cliques of $\Gamma_r^*({\mathcal{H}})$ are not equivalent under the action of ${\rm Aut}(\Gamma_r^*({\mathcal{H}}))$ and they fall into $\frac{1}{2}r+1$ and $\frac{1}{2}(r+3)$ classes according as $r$ is even or odd.
2015
Cossidente, A; Marino, Giuseppe; Korchm\`aros, G.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11591/170708
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