#####
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 version are available: