Graphs from Projective Planes

Abstract

The orthogonality relation among subspaces of a finite vector space is studied here by means of the corresponding graph. In the case we consider, this graph has some highly symmetric induced subgraphs. We find three infinite families of graphs of girth 3, and two infinite families of graphs of girth 5, whose automorphism groups are transitive on ordered pairs of adjacent points. Special cases are the Petersen graph, a 28-point cubic 3-transitive graph of girth 7 due to H. S. M. Coxeter, and a 36-point quintic 2-transitive graph of girth 5. The algebraic representations obtained for these graphs afford easy computation of their properties, and show that their automorphism groups are respectively the projective orthogonal groups PC/3 (5), P6 3 (7), and a semidirect product P~3 (9)# Z z. Our study is in the spirit of Coxeter's paper 'Self-dual configurations and regular graphs' [4]. In that paper, many interesting graphs were obtained as the Levi graphs of geometric configurations. A configuration (m c, ha) is a set ofm points and n lines in a plane, with dpoints on each line and c lines through each point. The configuration is self-dual if it has a duality (incidence-preserving bijection interchanging points and lines). The Levi graph is the red-blue bipartite graph joining each point (red vertex) to each line (blue vertex) incident to it in the configuration. This graph always has even girth. A polarity n is a duality such that 7~ 2 is the identity. We start from a configuration self-dual via a polarity n. Instead of the Levi graph, we use a 'polarity graph' whose vertices are the points X, Y,... of the configuration, and where X is adjacent to Y in the graph ifX-~ Yand X is on n(Y), the polar line of Y. Our configuration consists of the points and lines of the finite projective plane PG(2, q), where q is a prime power. The points and lines may be identified with the I-dimensional and 2-dimensional subspaces of the 3-dimensional vector space over the finite field F= GF(q). We let n (X) be the orthogonal complement of X. The resulting polarity graph is denoted by G(q).