On graphs with excess or defect 2

We prove that a graph with odd minimal degree, girth 9 and excess 2, i.e. a graph with two more vertices than the Moore bound, 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 girth higher than 9 and various 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. Amongst others we will prove that there is no graph with odd maximal degree, diameter 4 and defect 2. Together with previous results about the degree/diameter problem this completes the case of diameter 4 and defect 2.