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.
The following versions are available: