Séminaire Lotharingien de Combinatoire, B18e (1987), 2 pp.
[Formerly: Publ. I.R.M.A. Strasbourg, 1988, 358/S-18, p. 141.]

Gareth A. Jones

The Farey Graph

Abstract. The Farey graph F is the graph on the set of rational numbers, the infinity included, where the vertex infinity is joined to the integers, while two rational numbers r/s and x/y (in reduced form) are adjacent in F if and only if ry-sx=1 or -1, or equivalently if they are consecutive terms in some Farey sequence F(m) (consisting of the rationals x/y with |y|<= m, arranged in increasing order). We introduce generalizations of the Farey graph that arise in connection with the modular group PSL(2,Z) acting on this "augmented" set of rationals and investigate some of their properties.

This paper is a summary of:

G.A.Jones, D.Singerman and K.Wicks: The modular group and generalized Farey graphs, in Groups, St. Andrews 1989, vol. 2 (C.M.Campbell and E.F.Robertson eds.), London Math. Soc. Lecture Note Ser. 160 (1991), 316-338.

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