Edges in graphs with large girth

Abstract

Several upper bounds are given for the maximum number of edgese possible in a graph depending upon its orderp, girthg and, in certain cases, minimum degreeδ. In particular, one upper bound has an asymptotic order ofp ^1+2/(g−1) wheng is odd. A corollary of our final result is that$g \leqslant 2 + 2\log _k \left( {\frac{p}{4}} \right)$ whenk = ⌊e/p⌋ ≥ 2. Asymptotic and numerical comparisons are also presented.