The Cage Problem is one of the oldest problems in Graph Theory. The idea is to find smallest regular graphs with given degree and girth and with the minimum possible number of vertices. A regular graph is one in which every vertex has the same degree. The degree of a vertex is the number of edges incident with it, and the girth of a graph is the length of a shortest cycle. 
The links below point to files containing the smallest known graphs for certain values of degree and girth. Each graph is saved as an adjacency list. The vertices are labeled with the integers 0 to n1. Line 1 lists the neighbors of vertex 0, line 2 the neighbors of vertex 1, etc. Only my constructions are shown here, for more complete coverage see the Dynamic Survey. 

A page on the (3,14)graph 
A page on the (3,16)graph 
A page on the (3,17)graph 
A page on the (3,18)graph 
Recently, Robert Jajcay and I undertook an investigation of lifts of theta graphs, focusing on the use of SL(2,q) as the voltage group. The following record graphs were produced. 

Geoff Exoo