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

Abstract

A simple combinatorial construction is given which takes as its imput a regular graph of valency $\text{k}$ such that the convex closure of two points at distance two is the complete bipartite graph $\text{K}{}_{\mathrm{3,3}}$ and whose output is a regular graph of valency $\text{2k+1}$. It is shown that the sequence of graphs obtained by starting with the graph with one point and no edges and applying the construction recursively is the family of bipartite dual polar space of type $\text{DSO}{}^{\mathrm{+}}\text{(2n,2)}$.