CHAPTER 4 - SOME COMBINATORIAL APPLICATIONS OF FINITE GEOMETRIES

Abstract

This chapter highlights some combinatorial applications of finite geometries. By a graph, a figure consisting of points and of edges connecting the points, the following conditions are valid: (1) the graph has points, (2) an edge can join two distinct points only, and (3) two points are joined by at most one edge. If any two (distinct) points of a graph are joined by edges, it is said to be complete. The number of points of a finite graph is called the order of the graph, and the number of the edges will be called the class of the graph. A tree is a nonempty connected graph, which does not contain a cycle; hence, the girth of a tree is 0. If a cycle passes through every point of a graph, the cycle is called a Hamilton circuit and the graph a Hamilton graph. If the points of a graph can be subdivided into two nonempty classes so that an edge joins only points belonging to different classes, the graph will be called a bipartite graph of an even graph. Up to isomorphism, the Petersen graph is the only regular graph of degree three of minimal order whose smallest circuit consists of five edges. The incidence graph of a Galois plane is a regular, bipartite Hamilton graph of the minimal order, of degree (q+1), and of the girth 6.