Strongly regular graphs from hyperbolic quadrics and their maximal cliques
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| Format: | Preprint |
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2025
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| _version_ | 1866912886251585536 |
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| author | Cossidente, Antonio De Beule, Jan Marino, Giuseppe Pavese, Francesco Smaldore, Valentino |
| author_facet | Cossidente, Antonio De Beule, Jan Marino, Giuseppe Pavese, Francesco Smaldore, Valentino |
| contents | Let $Q^+(2n+1,q)$ be a hyperbolic quadric of $\PG(2n+1,q)$. Fix a generator $Π$ of the quadric. Define $\cG_n$ as the graph with vertex set the points of $Q^+(2n+1,q)\setminus Π$ and two vertices adjacent if they either span a secant to $Q^+(2n+1,q)$ or a line contained in $Q^+(2n+1,q)$ meeting $Π$ non-trivially. Then such a construction defines a strongly regular graph, which is the complement of a (non-induced) subgraph of the collinearity graph of $Q^+(2n+1,q)$. In this paper, we directly compute the parameters of $\cG_n$, which is cospectral, when $q=2$, to the tangent graph $NO^+(2n+2,2)$, but it is non-isomorphic for $n\geq3$. We also classify the maximal cliques of $\cG_3$ for $q=2$, proving as a by-product the non-isomorphism with the graph $NO^+(8,2)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_19560 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strongly regular graphs from hyperbolic quadrics and their maximal cliques Cossidente, Antonio De Beule, Jan Marino, Giuseppe Pavese, Francesco Smaldore, Valentino Combinatorics Primary 51E20, Secondary 05E30 Let $Q^+(2n+1,q)$ be a hyperbolic quadric of $\PG(2n+1,q)$. Fix a generator $Π$ of the quadric. Define $\cG_n$ as the graph with vertex set the points of $Q^+(2n+1,q)\setminus Π$ and two vertices adjacent if they either span a secant to $Q^+(2n+1,q)$ or a line contained in $Q^+(2n+1,q)$ meeting $Π$ non-trivially. Then such a construction defines a strongly regular graph, which is the complement of a (non-induced) subgraph of the collinearity graph of $Q^+(2n+1,q)$. In this paper, we directly compute the parameters of $\cG_n$, which is cospectral, when $q=2$, to the tangent graph $NO^+(2n+2,2)$, but it is non-isomorphic for $n\geq3$. We also classify the maximal cliques of $\cG_3$ for $q=2$, proving as a by-product the non-isomorphism with the graph $NO^+(8,2)$. |
| title | Strongly regular graphs from hyperbolic quadrics and their maximal cliques |
| topic | Combinatorics Primary 51E20, Secondary 05E30 |
| url | https://arxiv.org/abs/2504.19560 |