On bipartite graphs of defect 2

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 H D − 1 ( x ) ± 1 , where H D − 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 Q of rational numbers of the polynomial H D − 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 .