Independent sets in polarity graphs

TL;DR

This paper investigates the independence numbers of various polarity graphs derived from projective planes, extending known bounds from the Erdős-Rényi orthogonal polarity graph to more general families.

Contribution

It proves that eigenvalue methods accurately estimate the independence number for broader classes of polarity graphs beyond the orthogonal case.

Findings

Eigenvalue bounds match the independence number order of magnitude for new polarity graph families.

The paper extends known results from Erdős-Rényi graphs to more general polarity graphs.

Conjectures that all polarity graphs of a projective plane have large independent sets of size (q^{3/2}).

Abstract

Given a projective plane $\Sigma$ and a polarity $\theta$ of $\Sigma$, the corresponding polarity graph is the graph whose vertices are the points of $\Sigma$, and two distinct points $p_1$ and $p_2$ are adjacent if $p_1$ is incident to $p_2^{ \theta}$ in $\Sigma$. A well-known example of a polarity graph is the Erd\H{o}s-R\'{e}nyi orthogonal polarity graph $ER_q$, which appears frequently in a variety of extremal problems. Eigenvalue methods provide an upper bound on the independence number of any polarity graph. Mubayi and Williford showed that in the case of $ER_q$, the eigenvalue method gives the correct upper bound in order of magnitude. We prove that this is also true for other families of polarity graphs. This includes a family of polarity graphs for which the polarity is neither orthogonal nor unitary. We conjecture that any polarity graph of a projective plane of order $q$ has an independent set of size $\Omega (q^{3/2})$. Some related results are also obtained.