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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.