Self-dual configurations and their Levi graphs

Abstract

Introduction. By a configuration we shall mean a set of points and straight lines (or planes) between which certain well-determined incidences exist. These configurations are not necessarily regular, that is, we shall not suppose every line (plane) to be incident with the same number of configuration points, and dually, different points of the configuration need not be incident with the same number of configuration lines (planes). The present paper will treat only self-dual configurations. (In this section and the next, we shall speak of configurations of points and lines only, but the reader may put plane instead of line.) As Coxeter [I ] has pointed out, configurations have been described by graphs of two kinds: the Menger graph which has the disadvantage of not determining uniquely the configuration which it describes, and the Levi graph which in general is rather cumbersome because it has one node for each point and another for each line of the configuration. Thus the Levi graph of a self-dual configuration of n points has 2n nodes. In order to overcome this inconvenience of the Levi graph we propose here to identify abstractly the node of every point with the node of its dually corresponding line, and thus to construct a reduced Levi graph (in short RLG) which has only n nodes and which preserves the advantages of the Levi graph: one RLG cannot describe two distinct configurations. On the other hand the RLG will be shown to have the disadvantage that one configuration may have several nonisomorphic RLG's. In RLG's a node may be joined to itself, thus forming a loop. We shall see that the RLG permits sometimes a convenient discussion of self-dual specializations of self-dual configurations, in addition to its uses as a simpler form of the Levi graph.