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