In this paper we give a method for obtaining the adjacency matrix of a simple polarity graph Gq from a projective plane $PG(2, q)$, where $q$ is a prime power. Denote by $ex(n;C_4)$ the maximum number of edges of a graph on $n$ vertices and free of squares $C_4$. We use the constructed graphs $G_q$ to obtain lower bounds on the extremal function $ex(n;C_4)$, for some $n < q^2 + q + 1$. In particular, we construct a $C_4$--free graph on $n = q^2 − \sqrt{q}$ vertices and $\frac{1}{2}q(q^2 − 1) − \frac{1}{2} \sqrt{q}(q − 1)$ edges, for a square prime power $q$.
Adjacency matrices of polarity graphs and other C_4-free graphs of large size
ABREU, Marien;LABBATE, Domenico
2010-01-01
Abstract
In this paper we give a method for obtaining the adjacency matrix of a simple polarity graph Gq from a projective plane $PG(2, q)$, where $q$ is a prime power. Denote by $ex(n;C_4)$ the maximum number of edges of a graph on $n$ vertices and free of squares $C_4$. We use the constructed graphs $G_q$ to obtain lower bounds on the extremal function $ex(n;C_4)$, for some $n < q^2 + q + 1$. In particular, we construct a $C_4$--free graph on $n = q^2 − \sqrt{q}$ vertices and $\frac{1}{2}q(q^2 − 1) − \frac{1}{2} \sqrt{q}(q − 1)$ edges, for a square prime power $q$.File in questo prodotto:
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