Elliptic semiplanes and regular graphs with girth 5

A (k, g)-graph is a k-regular graph with girth g and a (k, g)-cage is a (k, g)-graph with the fewest possible number of vertices. The cage problem consists of constructing (k, g)-graphs of minimum order n(k, g). We focus on girth g=5, where cages are known only for degrees k≤7. Considering the relationship between finite geometries and graphs we establish upper constructive bounds on n(k, 5), for k∈{13,14,17,18,…} that improve the best so far known.