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 Δ≥2Δ≥2, diameter D≥2D≥2 and defect 2 (having 2 vertices less than the Moore bipartite bound). We call such graphs bipartite (Δ,D,−2)(Δ,D,−2)-graphs.We find that the eigenvalues other than ±Δ±Δ of such graphs are the roots of the polynomials HD−1(x)±1HD−1(x)±1, where HD−1(x)HD−1(x) is the Dickson polynomial of the second kind with parameter Δ−1Δ−1 and degree D−1D−1.For any diameter, we prove that the irreducibility over the field QQ of rational numbers of the polynomial HD−1(x)−1HD−1(x)−1 provides a sufficient condition for the non-existence of bipartite (Δ,D,−2)(Δ,D,−2)-graphs for Δ≥3Δ≥3 and D≥4D≥4. Then, by checking the irreducibility of these polynomials, we prove the non-existence of bipartite (Δ,D,−2)(Δ,D,−2)-graphs for all Δ≥3Δ≥3 and D∈{4,6,8}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)(Δ,5,−2)-graphs for all Δ≥3Δ≥3.Finally, we conjecture that there are no bipartite (Δ,D,−2)(Δ,D,−2)-graphs for Δ≥3Δ≥3 and D≥4D≥4.