On almost nexus semi-symmetric designs (Q1092052)

A semi-symmetric design (SSD) is a finite connected incidence structure satisfying (i) two points are in 0 or \(\lambda\) blocks, and dually; (ii) each block contains \(k>\lambda\) points, and each point is on \(r>\lambda\) blocks. In case \(\lambda\neq 1\), one gets \(b=v\), \(r=k\); in case \(\lambda =1\), this is assumed to hold. An SSD has the almost nexus property if for any point \(p^{\alpha}\) there exists a unique block \(p^{\alpha}\) with \(p\not\in p^{\alpha}\) such that the number of points on \(p^{\alpha}\) joined to p is a constant e whereas for any block \(Y\neq p^{\alpha}\) with \(p\not\in Y\) the number of points on Y joined to p is a constant \(s>\lambda\). The author proves the following results: If \(e\neq s\), then \(\alpha\) is a polarity; this gives some divisibility conditions for the parameters. In case \(e=0\) or \(e=k\), the SSD gives rise to a distance regular graph; in case \(e=0\) or \(e=k>s\) such an SSD is actually equivalent to a certain distance regular graph. If \(\lambda\) is square- free, \(e=0\) and \(k>s+\lambda\), then \(v=57\), \(k=6\), \(s=3\), \(\lambda =1\). If \(\lambda\) is square-free and \(e=k>s\), then \(v=21\), \(k=4\), \(s=2\), \(\lambda =1\).