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

TL;DR

This paper investigates the vertex-isoperimetric number of the incidence graph of projective spaces, providing asymptotic estimates and exact values for specific cases, advancing understanding of combinatorial properties of these graphs.

Contribution

It determines the order of magnitude of the vertex-isoperimetric number for the incidence graph of PG(n,q) and computes exact values for PG(2,q) with q ≤ 16.

Findings

Asymptotic order of the vertex-isoperimetric number for PG(n,q)

Exact values of the vertex-isoperimetric number for PG(2,q) with q ≤ 16

Results contribute to combinatorial understanding of incidence graphs

Abstract

Let $\Gamma_{n,q}$ be the point-hyperplane incidence graph of the projective space $\operatorname{PG}(n,q)$, 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 $q \le 16$.