The regular near polygons of order \((s,2)\) (Q1768528)

A near polygon is a partial linear space \({\mathcal S}=({\mathcal P},{\mathcal L})\) with the property that for every point \(p\in {\mathcal P}\) and for every line \(L\in {\mathcal L}\) there exists a unique point on \(L\) nearest to \(p\). Here distances \(d\) are measured in the collinearity graph \(\Gamma\). If \(n=\text{ diam}(\Gamma)\), then \({\mathcal S}\) is called a near \(2n\)-gon. It is well known that (the collinearity graph \(\Gamma\) of) a regular near polygon of order \((s,t)\) is a distance-regular graph of valency \(s(t+1)\), diameter \(d\) and \(a_i=c_i(s-1)\) for all \(1\leq i\leq d-1\) such that for any vertex \(x\) the subgraph \(\Gamma(x)\) is the disjoint union of \(t+1\) cliques of size \(s\). Theorem 1. A regular near polygon of order \((s,2)\) is isomorphic to \(K_{3,3}\), the Petersen graph, the Heawood graph (\(v=14\)), the Pappus graph (\(v=18\)), Tutte's 8 cage (\(v=30\)), the Desargues graph (\(v=20\)), Tutte's 12 cage (\(v=126\)), the Foster graph (\(v=90\)), the generalized quadrangle GQ\((2,2)\) or GQ\((4,2)\), the generalized hexagon GH\((2,2)\) or GQ\((8,2)\), the generalized octagon GO\((4,2)\) or the Hamming graph \(H(3,s+1)\).