On bipartite graphs of defect 2

  • Authors:

  • Charles Delorme;Leif K. Jørgensen;Mirka Miller;Guillermo Pineda-Villavicencio

  • Affiliations:

  • Lab. de Recherche en Informatique, University de Paris-Sud, 91405 Orsay, France;Department of Mathematics and Computer Science, Aalborg University, F. Bajers Vej 7, DK-9220 Aalborg Ø, Denmark;School of Electrical Engineering and Computer Science, The University of Newcastle, Callaghan, New South Wales 2308, Australia and Department of Mathematics, University of West Bohemia, Univerzitn ...;School of Information Technology and Mathematical Sciences, University of Ballarat, Mount Helen, Victoria 3353, Australia

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2009

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Abstract

It is known that the Moore bipartite bound provides an upper bound on the order of a connected bipartite graph. In this paper we deal with bipartite graphs of maximum degree Δ≥2, diameter D≥2 and defect 2 (having 2 vertices less than the Moore bipartite bound). We call such graphs bipartite (Δ,D,-2)-graphs. We find that the eigenvalues other than ±Δ of such graphs are the roots of the polynomials HD-1(x)±1, where HD-1(x) is the Dickson polynomial of the second kind with parameter Δ-1 and degree D-1. For any diameter, we prove that the irreducibility over the field of rational numbers of the polynomial HD-1(x)-1 provides a sufficient condition for the non-existence of bipartite (Δ,D,-2)-graphs for Δ≥3 and D≥4. Then, by checking the irreducibility of these polynomials, we prove the non-existence of bipartite (Δ,D,-2)-graphs for all Δ≥3 and D∈{4,6,8}. For odd diameters, we develop an approach that allows us to consider only one partite set of the graph in order to study the non-existence of the graph. Based on this, we prove the non-existence of bipartite (Δ,5,-2)-graphs for all Δ≥3. Finally, we conjecture that there are no bipartite (Δ,D,-2)-graphs for Δ≥3 and D≥4.