Explicit construction of graphs with an arbitrary large girth and of large size Original Research Article

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 = 2qk 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 = 2t-regular bipartite graph on v = 2qk + 1 vertices with girth g ⩾ k + 5 which is both vertex-transitive and edge-transitive.