A New Series of Dense Graphs of High Girth

Let k ≥ 1 be an odd integer, t = ⌊ k+2 ⌋, and q be a prime power. We 4 construct a bipartite, q–regular, edge–transitive graph CD(k, q) of order v ≤ 2q k−t+1 and girth g ≥ k + 5. If e is the the number of edges of CD(k, q), then e = Ω(v 1+ 1 k−t+1). These graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order v and girth at least g, g ≥ 5, g � = 11, 12. For g ≥ 24, this represents a slight improvement on bounds established by Margulis and Lubotzky, Phillips, Sarnak; for 5 ≤ g ≤ 23, g � = 11, 12, it improves on or ties existing bounds