Explicit Construction of Graphs with an Arbitrary Large Girth and of Large Size

Let k ≥ 3 be a positive odd integer and q be a power of a prime. In this paper we give an explicit construction of a q–regular bipartite graph on v = 2q vertices with girth g ≥ k + 5. The constructed graph is the incidence graph of a flag–transitive semiplane. For any positive integer t we also give an example of a q = 2–regular bipartite graph on v = 2q vertices with girth g ≥ k + 5 which is both vertex–transitive and edge–transitive . Section