The isoperimetric number of the incidence graph of PG(n,q)

Abstract

Let ${\Gamma }_{n,q}$ be the point-hyperplane incidence graph of the projective space $\mathrm{PG}(n,q)$, where $n\ge 2$ is an integer and $q$ a prime power. We determine the order of magnitude of $1-i_V({\Gamma }_{n,q})$, where $i_V({\Gamma }_{n,q})$ is the vertex-isoperimetric number of ${\Gamma }_{n,q}$. We also obtain the exact values of $i_V({\Gamma }_{2,q})$ and the related incidence-free number of ${\Gamma }_{2,q}$ for $q\le 16$.