Bounds for graphs of given girth and generalized polygons

Lakdere Benkherouf ; Vasyl Ustimenko

Algebra and discrete mathematics, no. 1 (2002), pp. 1-18.

Résumé

In this paper we present a bound for bipartite graphs with average bidegrees $\eta$ and $\xi$ satisfying the inequality $\eta \ge {\xi }^{\alpha }$, $\alpha \ge 1$. This bound turns out to be the sharpest existing bound. Sizes of known families of finite generalized polygons are exactly on that bound. Finally, we present lower bounds for the numbers of points and lines of biregular graphs (tactical configurations) in terms of their bidegrees. We prove that finite generalized polygons have smallest possible order among tactical configuration of given bidegrees and girth. We also present an upper bound on the size of graphs of girth $g\ge 2t+1$. This bound has the same magnitude as that of Erdös bound, which estimates the size of graphs without cycles $C_{2t}$.

Détail

Mots-clés : Extremal Graph Theory, Operations Research, Family of Graphs of High Girth, Simple Groups of Lie Type.