Abstract
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.
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