Let Q+ (2n+1, q) be a hyperbolic quadric of PG(2n+1, q). Fix a generator Π of the quadric. Define Gn as the graph with vertex set the points of Q+ (2n + 1, q) \ 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 Gn, which is cospectral, when q = 2, to the tangent graph NO+ (2n + 2, 2), but it is non-isomorphic for n ≥ 3. We also classify the maximal cliques of G3 for q = 2, proving as a by-product the non-isomorphism with the graph NO+ (8, 2).
Strongly regular graphs from hyperbolic quadrics and their maximal cliques
Valentino Smaldore
2026
Abstract
Let Q+ (2n+1, q) be a hyperbolic quadric of PG(2n+1, q). Fix a generator Π of the quadric. Define Gn as the graph with vertex set the points of Q+ (2n + 1, q) \ 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 Gn, which is cospectral, when q = 2, to the tangent graph NO+ (2n + 2, 2), but it is non-isomorphic for n ≥ 3. We also classify the maximal cliques of G3 for q = 2, proving as a by-product the non-isomorphism with the graph NO+ (8, 2).
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