New family of small regular graphs of girth 5

Abstract

A (k,g)-cage is a k-regular graph of girth g of minimum order. In this work, we focus on girth g=5, where cages are known only for degrees k≤7. When k≥8, except perhaps for k=57, the order of a (k,5)-cage is strictly greater than 1+k2. Considering the relationship between finite geometries and graphs we establish upper constructive bounds that improve the best so far.