On graphs with excess or defect 2

The Moore bound m(d, k) = 1 + d ∑k−1 i=0 (d − 1)i is a lower bound for the number of vertices of a graph by given girth g = 2k + 1 and minimal degree d. Hoffmann and Singleton [5], Bannai and Ito [1], Damerell [4] showed that graphs with d > 2 tight to this bound can only exist for girth 5 and degree 3, 7, 57. The difference to the Moore bound by given girth is called the excess of a graph. In the case of girth 5 Brown showed in [3] that there are no graphs with excess 1 and Bannai and Ito showed in [2] that for g ≥ 7 there are also no graphs with excess 1. We generalize the result of Kovács [6] that, under special conditions for the degree d, there are no graphs with excess 2 and girth 5 and prove the new result that an excess-2-graph with odd degree and girth 9 cannot exist. In this proof we discover a link to certain elliptic curves. Furthermore we prove the non-existence of graphs with excess 2 for higher girth and special valencies under certain congruence conditions. The results can be modified to fit the degree/diameter problem and lead to similar statements for graphs with defect 2.