3-regularity in generalized quadrangles of order \((s,s^ 2)\)
Let \(S=(P,B,I)\) be a generalized quadrangle (GQ) of order \((s,s^ 2)\), \(s>1\). Then for any triad (x,y,z), i.e. for any triple (x,y,z) of pairwise noncollinear points, we have \(| \{x,y,z\}^{\perp}| =s+1.\) We say that the triad (x,y,z) is 3-regular provided \(| \{x,y,z\}^{\perp \perp}| =s+1.\) Let (x,y,z) be a 3-regular triad of the GQ S\(=(P,B,I)\) of order \((s,s^ 2)\), \(s>1\), and let P' be the set of all points incident with lines of the form uv, \(u\in \{x,y,z\}^{\perp}=X\) and \(v\in \{x,y,z\}^{\perp \perp}=Y.\) If L is a line which is incident with no point of \(X\cup Y\) and if k is the number of points in P' which are incident with L, then it is shown that \(k\in \{0,2\}\) if s is odd and \(k\in \{1,s+1\}\) if s is even. If s is even, and if B' is the set of all lines in B which are incident with at least two points of P', then it is proved that \(S'=(P',B',I')\) with \(I'=I\cap (P'\times B')\) is a subquadrangle of order s. By applying that results we show that any GQ of order (4,16) containing a 3-regular triad is isomorphic to Q(5,4), the GQ arising from the elliptic quadric of PG(5,4). Corollary: Let \(\Gamma\) be a strongly regular graph on 325 vertices with \(k=68\), \(\lambda =3\) and \(\mu =17\); if \(\Gamma\) contains a complete bipartite graph \(K_{5,5}\) as an induced subgraph, then \(\Gamma\) is the point graph of the GQ Q(5,4).
scientific article; zbMATH DE number 3888425