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

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; C4) the maximum number of edges of a graph on n vertices and free of squares C4. We use the constructed graphs Gq 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 $${n=q^2 -\sqrt{q}}$$ vertices and $${\frac{1}{2} q(q^2-1)-\frac{1}{2}\sqrt{q} (q-1) }$$ edges, for a square prime power q.

[1]  Mehdi Behzad,et al.  Graphs and Digraphs , 1981, The Mathematical Gazette.

[2]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[3]  Pak-Ken Wong,et al.  Cages - a survey , 1982, J. Graph Theory.

[4]  W. G. Brown.  On Graphs that do not Contain a Thomsen Graph , 1966, Canadian Mathematical Bulletin.

[5]  A. Rényii,et al.  ON A PROBLEM OF GRAPH THEORY , 1966 .

[6]  Reinhard Laue,et al.  Resolvable t-Designs , 2004, Des. Codes Cryptogr..

[7]  Paul Erdös,et al.  Graphs of diameter two with no 4-circuits , 1999, Discret. Math..

[8]  William M. Kantor.  MOORE GEOMETRIES AND RANK 3 GROUPS HAVING μ=1 , 1977 .

[9]  G. Exoo,et al.  Dynamic Cage Survey , 2011 .

[10]  Zoltán Füredi,et al.  Graphs without quadrilaterals , 1983, J. Comb. Theory, Ser. B.

[11]  C. R. J. Clapham,et al.  Graphs without four-cycles , 1989, J. Graph Theory.

[12]  Camino Balbuena,et al.  Incidence Matrices of Projective Planes and of Some Regular Bipartite Graphs of Girth 6 with Few Vertices , 2008, SIAM J. Discret. Math..

[13]  R. Baer.  Polarities in finite projective planes , 1946 .

[14]  Zoltán Füredi,et al.  On the Number of Edges of Quadrilateral-Free Graphs , 1996, J. Comb. Theory, Ser. B.

[15]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[16]  L. Ji,et al.  Resolvable Steiner Quadruple Systems for the Last 23 Orders , 2005, SIAM J. Discret. Math..