An infinite family of locally X graphs based on incidence geometries

Abstract

A graph $\mathcal{G}$ is locally X if the graphs induced on the neighbours of every vertex of $\mathcal{G}$ are isomorphic to the graph $X$. We prove that the infinite family of incidence graphs of the $r$-rank incidence geometries, $\Gamma (KG(n,k),r)$, constructed using the Kneser graphs $KG(n,k)$, are locally $X$ with $X$ being the incidence graphs of the rank $r-1$ residues of $\Gamma (KG(n,k),r)$.