Generalized polygons are rank 2 geometries that were introduced by Jacques Tits in order to better understand the twisted triality groups, see [29]. As pre-cursors of buildings, they were the spherical rank 2 buildings avant-la-lettre. The standard examples are related to simple algebraic groups

$. A generahzed polygon is a bipartite graph with girth equal to twice the diameter. A generalized polygon of diameter $n $ is also called a generalized n-gon or generalized triangle for $n=3 $ , quadrangle for $n=4$, etc. A generalized 2-gon is just a complete bipartite graph. Suppose $\triangle

Recently there has been a great deal of activity in the geometric and group theoretic study of "building-like geometries" One of the directions of this activity has concerned GABs ("geometries that are almost buildings") and SCABs ("chamber systems that are almost buildings"). (Other names for these are "chamber systems of type M" and "geometries of type M" in [Ti 7], and "Tits geometries of type M" or "Tits chamber systems of type M" in [AS; Tim 1,2]. ) The main goal of this paper is to survey these developments with special emphasis on their relationships with finite geometries.

While the theory and applications of discrete Laplacians on triangulated surfaces are well developed, far less is known about the general polygonal case. We present here a principled approach for constructing geometric discrete Laplacians on surfaces with arbitrary polygonal faces, encompassing non

Abstract not found

and conversely

We introduce stable graphs as a common generalization of compact generalized poly-gons with closed adjacency, stable planes and other types of graphs with continuous ge-ometric operations; non-bipartite structures like Moore graphs are also included. Topo-logical and graph-theoretical properties

: A novel type of skeleton for general polygonal figures, the straight skeleton S(G) of a planar straight line graph G, is introduced and discussed. Exact bounds on the size of S(G) are derived. The straight line structure of S(G) and its lower combinatorial complexity may make S(G) preferable

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

Abstract. This paper presents fast recursive or moving windows algorithms for calculating local means in a diamond, hexagon and general polygonal shaped windows of an image. The algorithms for diamond shaped window require only seven or eight additions and subtractions per pixel. A number of other