Homogeneous graphs and regular near polygons (Q1322008)

A homogeneous graph \(\Gamma\) is defined: for every edge \(uv\), and vertex \(x\), the number of edges from \(x\) to \(\Gamma_ i(u)\cap\Gamma_ j(v)\) depends only on \(i\), \(j\) and the distances from \(x\) to \(u\) and \(v\). (\(\Gamma_ i(u)\) is the set of vertices of distance \(i\) from \(u\).) It is proven that, for distance-regular graphs in which the set of common neighbors of adjacent vertices is always a clique, homogeneous graphs are precisely the regular near \(2d\)-gons.