On bipartite graphs of defect 2

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 ∆ ≥ 2, diameter D ≥ 2 and defect 2 (having 2 vertices less than the Moore bipartite bound). We call such graphs bipartite (∆, D,−2)-graphs. We find that the eigenvalues other than ±∆ 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 ∆− 1 and degree D − 1. For any diameter, we prove that the irreducibility over the field Q of rational numbers of the polynomial HD−1(x)−1 provides a sufficient condition for the non-existence of bipartite (∆, D,−2)-graphs for ∆ ≥ 3 and D ≥ 4. Then, by checking the irreducibility of these polynomials, we prove the non-existence of bipartite (∆, D,−2)-graphs for all ∆ ≥ 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 (∆, 5,−2)-graphs for all ∆ ≥ 3. Finally, we conjecture that there are no bipartite (∆, D,−2)-graphs for ∆ ≥ 3 and D ≥ 4.