https://ogma.newcastle.edu.au/vital/access/ /manager/Index ${session.getAttribute("locale")} 5 https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:9198 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 ≥ 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,−ε)-graphs with D ≥ 2 and 0 ≤ ε ≤ 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 ≥ 2 and 0 ≤ ε ≤ 2 [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 (mod D). 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).]]> Wed 11 Apr 2018 12:46:35 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:17695 Wed 11 Apr 2018 11:41:51 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:4975 Wed 11 Apr 2018 09:35:56 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:7828 Sat 24 Mar 2018 08:37:39 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:5916 Sat 24 Mar 2018 07:46:46 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:68 Sat 24 Mar 2018 07:42:02 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:29677 Moore bound) are extremely rare, but much activity is focused on finding new examples of graphs or families of graph with orders approaching the bound as closely as possible. There has been recent interest in this problem as it applies to mixed graphs, in which we allow some of the edges to be undirected and some directed. A 2008 paper of Nguyen and Miller derived an upper bound on the possible number of vertices of such graphs. We show that for diameters larger than three, this bound can be reduced and we present a corrected Moore bound for mixed graphs, valid for all diameters and for all combinations of undirected and directed degrees.]]> Sat 24 Mar 2018 07:32:21 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:38494 Mon 11 Oct 2021 14:41:00 AEDT ]]>