Explicit construction of graphs with an arbitrary large girth and of large size

Let k 3 be a positive odd integer and 1 be a power of a prime. In this paper we give an explicit construction of a q-regular bipartite graph on v = 2q^k 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^t-regular bipartite graph on v = 2q^{k + 1} vertices with girth g k + 5 which is both vertex-transitive and edge-transitive.