Girth 5 graphs from relative difference sets

We consider the problem of construction of graphs with given degree k and girth 5 and as few vertices as possible. We give a construction of a family of girth 5 graphs based on relative difference sets. This family contains the smallest known graph of degree 8 and girth 5 which was constructed by Royle, four of the known cages including the Hoffman-Singleton graph, some graphs constructed by Exoo and some new smallest known graphs. (c) 2005 Elsevier B.V. All rights reserved.