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Adjacency matrices of polarity graphs and of other C 4-free graphs of large size

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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; 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.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi della Basilicata, Viale dell’Ateneo Lucano, 85100, Potenza, Italy

    M. Abreu

  2. Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Campus Nord, Edifici C2, C/ Jordi Girona 1 i 3, 08034, Barcelona, Spain

    C. Balbuena

  3. Dipartimento di Matematica, Politecnico di Bari, 70125, Bari, Italy

    D. Labbate

Authors
  1. M. Abreu
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  2. C. Balbuena
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  3. D. Labbate
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Correspondence to D. Labbate.

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Communicated by Ron Mullin, Rainer Steinwandt.

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Abreu, M., Balbuena, C. & Labbate, D. Adjacency matrices of polarity graphs and of other C 4-free graphs of large size. Des. Codes Cryptogr. 55, 221–233 (2010). https://doi.org/10.1007/s10623-010-9364-1

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  • DOI: https://doi.org/10.1007/s10623-010-9364-1

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