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On graphs of defect at most 2

In this paper we consider the degree/diameter problem, namely, given natural numbers Δ \geq 2 and D \geq 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, turned out to be very rare. Therefore, it is very interesting to investigate graphs of maximum degree Δ \geq 2, diameter D \geq 1 and order M(Δ,D) - ε with small ε > 0, that is, (Δ,D,-ε)-graphs. The parameter ε is called the defect. Graphs of defect 1 exist only for Δ = 2. When ε > 1, (Δ,D,-ε)-graphs represent a wide unexplored area. This paper focuses on graphs of defect 2. Building on the approaches developed in [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 \geq 4 and D \geq 4 is 2D. Second, and most important, we prove the non-existence of (Δ,D,-2)-graphs with even Δ \geq 4 and D \geq 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,-ε)-graphs with D \geq 2 and 0 \leq ε \leq 2. Such a catalogue is only the second census of (Δ,D,-2)-graphs known at present, the first being the one of (3,D,-ε)-graphs with D \geq 2 and 0 \leq ε \leq 2 [14]. Other results of this paper include necessary conditions for the existence of (Δ,D,-2)-graphs with odd Δ \geq 5 and D \geq 4, and the non-existence of (Δ,D,-2)-graphs with odd Δ \geq 5 and D \geq 5 such that Δ \equiv 0, 2 (mod D).