On graphs of defect at most 2

In this paper we consider the degree/diameter problem, namely, given natural numbers Δ<2 and D<1, find the maximum number N(Δ,D) of vertices in a graph of maximum degree Δ and diameter D. In this context, the Moore bound M(Δ,D) represents an upper bound for N(Δ,D). Graphs of maximum degree Δ, diameter D and order M(Δ,D), called Moore graphs, have turned out to be very rare. Therefore, it is very interesting to investigate graphs of maximum degree Δ<2, diameter D<1 and order M(Δ,D)- with small >0, that is, (Δ,D,-)-graphs. The parameter is called the defect. Graphs of defect 1 exist only for Δ=2. When >1, (Δ,D,-)-graphs represent a wide unexplored area. This paper focuses on graphs of defect 2. Building on the approaches developed in Feria-Purón and Pineda-Villavicencio (2010) [11] we obtain several new important results on this family of graphs. First, we prove that the girth of a (Δ,D,-2)-graph with Δ<4 and D<4 is 2D. Second, and most important, we prove the non-existence of (Δ,D,-2)-graphs with even Δ<4 and D<4; this outcome, together with a proof on the non-existence of (4,3,-2)-graphs (also provided in the paper), allows us to complete the catalogue of (4,D,-)-graphs with D<2 and 0≤≤2. Such a catalogue is only the second census of (Δ,D,-2)-graphs known at present, the first being that of (3,D,-)-graphs with D<2 and 0≤≤2 Jørgensen (1992) [14]. Other results of this paper include necessary conditions for the existence of (Δ,D,-2)-graphs with odd Δ<5 and D<4, and the non-existence of (Δ,D,-2)-graphs with odd Δ<5 and D<5 such that Δ≡0,2(modD). Finally, we conjecture that there are no (Δ,D,-2)-graphs with Δ<4 and D<4, and comment on some implications of our results for the upper bounds of N(Δ,D).