Adjacency matrices of polarity graphs and of other C4-free graphs of large size

Abstract

In this paper we give a method for obtaining the adjacency matrix of a simple polarity graph $G_q$ from a projective plane PG(2,q), where q is a prime power. Denote by ex(n;C₄) the maximum number of edges of a graph on n vertices and free of squares C₄. We use the constructed graphs $G_q$ to obtain lower bounds on the extremal function ex(n;C₄), for some n < q² + q + 1. In particular, we construct a C₄-free graph on n = q² − √q vertices and 1/2q(q²−1) − 1/2√q(q − 1) edges, for a square prime power q.