THE WRITHE OF ORIENTED POLYGONAL GRAPHS

Given an edge-oriented polygonal graph in ℝ3, we describe a method for computing the writhe as the average of weighted directional writhe numbers of the graph in a few directions. These directions are determined by the graph and the weights are determined by areas of path-connected open regions on the unit sphere. Within each open region, the directional writhe is constant. We obtain a closed formula which extends the formula for the writhe of a polygon in ℝ3, including the important special case of writhe of embedded open arcs.

A Taxonomic Research Update of Cultivated Liriopogons

A taxonomic revision of liriopogons (Liriope Lour., Ophiopogon Ker-Gawl) cultivated in the United States is in progress at North Carolina State Univ. Germplasm was obtained from nurseries, botanical gardens/arboreta, and private collectors. Nearly 17% of the germplasm was misidentified to genus; nearly 36% misidentified to species; and nearly 14% received under one name from one source contained mixed germplasm. Preliminary analysis of data indicate a minimum of five species of Liriope and eight species of Ophio-pogon are in cultivation. Six additional taxa have not flowered. Polygonal graph analysis was used to visualize biometrical data and observe relationships among taxa. Additional taxonomic publications for segregation of genera and species of liriopogons, including an inventory of taxa, quantitative descriptions, illustrations, and keys, are in progress.

A Characterization of PSL(2, 31) and its Geometry

The aim of this paper is the characterization of PSL(2, 31) in terms of its action on a certain polygonal graph. A polygonal graph is a pair (, ) consisting of a graph which is regular, connected and has girth m for some m ≧ 3, and a set of m-gons (circuits of length m) of such that every 2-claw (i.e. path of length 2) of is contained in a unique element of , (See Section 2 for the definitions of the terms used here.) If is the set of all m-gons of H, so that there is in a unique m-gon on every one of its 2-claws, then we write for (, ) and call a strict polygonal graph. If we wish to emphasize the integer m, then we call (, ) an m-gon-graph (respectively, a strict m-gon-graph). For convenience, a strict 5-gon-graph will be called a pentagraph.

Bounding the Valency of Polygonal Graphs With Odd Girth

In this paper we investigate the action of finite groups G on finite polygonal graphs. The notion of a polygonal graph was introduced in [17]: A polygonal graph is a pair (, ) consisting of a graph which is regular, connected and has girth m for some m ≧ 3, and a set of m-gons of such that every 2-claw of is contained in an unique element of (See Section 2 for the définitions of the terms used here.) If is the set of all m-gons of , so that there is in an unique m-gon on every one of its 2-claws, then we write for (, ) and call a strict polygonal graph. If we wish to emphasize the integer m, then we call (, ) an m-gon-graph (respectively, a strict m-gon-graph).