A combinatorial construction of a graph with automorphism group \(\text{SO}^+ (2n,2)\) (Q1567658)

This paper is concerned with a class of bipartite regular graphs with automorphism group \(\text{SO}^+ (2n,2)\), namely the collinearity graphs of the dual polar spaces of type \(\text{DSO}^+ (2n,2)\) arising from a hyperbolic quadric in dimension \(2n\) over the field of order \(2\). It is shown that these graphs can be constructed in a purely combinatorial fashion without any knowledge of orthogonal geometry, by first describing a construction process and then proving that if the construction is recursively applied, the family of graphs obtained by starting with a single vertex is isomorphic to the family \(\text{DSO}^+ (2n,2)\). The combinatorial construction takes as its input a \(k\)-regular graph with the property that the convex closure of two points at distance two is \(K_{3,3}\). The output is a \((2k+1)\)-regular graph. If the input is what the author calls a classical graph, for example a near \(2n\)-gon, the construction yields a graph inheriting the convex closure property specified for the input.