EXPLICIT CONSTRUCTION OF GRAPHS WITH AN ARBITRARY LARGE GIRTH AND OF LARGE-SIZE

Let k greater than or equal to 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(k) vertices with girth g greater than or equal to 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(k+1) vertices with girth g greater than or equal to k + 5 which is both vertex-transitive and edge-transitive.