Bounds for graphs of given girth and generalized polygons

• Опубліковано в: : Algebra and Discrete Mathematics
• Дата: 2002
• Автори: Benkherouf, L., Ustimenko, V.
• Формат: Стаття
• Мова: Англійська
• Опубліковано: Інститут прикладної математики і механіки НАН України 2002
• Онлайн доступ: https://nasplib.isofts.kiev.ua/handle/123456789/154677
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: Bounds for graphs of given girth and generalized polygons / L. Benkherouf, V. Ustimenko // Algebra and Discrete Mathematics. — 2002. — Vol. 1, № 1. — С. 1–18. — Бібліогр.: 26 назв. — англ.

In this paper we present a bound for bipartite
 graphs with average bidegrees η and ξ satisfying the inequality η ≥ ξ
 α, α ≥ 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 ≥ 2t + 1. This bound has the same magnitude
 as that of Erd¨os bound, which estimates the size of graphs without
 cycles C₂t.