4
ALESSANDRO FIGA-TALAMANCA
define a (nonoriented) graph (X
,E).
More conditions are required for a graph to
be a tree.
We can define a path in the graph as a finite sequence of vertices
xo, ... Xn
such
that the two-elements sets
{Xi,
xi+
1
}
belong to IE. A graph is connected if any pair
of vertices may be joined by a path, in other words if given x, y
E
X, there exists a
path
xo, ... Xn,
with
x
=
Xo
andy=
Xn·
The condition that (X,
E)
has no circuits simply means that if a path satisfies
the additional hypothesis that
Xi
#
xi+2 (that is there is "no return") it cannot
happen that
xo
=
Xn.
For the record, a path satifying the condition
Xi
#
Xi+2
is
called a chain. Thus a connected graph is a tree if there is only one chain connecting
two given vertices.
We introduce now the notation [x,
y]
to denote the unique chain joining the
vertices x, y
E
X.
Even though there is a lot of harmonic analysis which can be done on rather
general trees, in these lectures I will consider only trees which are locally finite and
homogeneous.
p-4o
p-3o
p-2o
H-2
p-lo
H-1
0
Ho
pO H1
p20 H2
Fig.1
A tree is called locally finite if each vertex belongs to a finite number of edges.
It
is homogeneous if this number (called the degree of the vertex) is the same for
every vertex. We shall talk about homogeneous trees of degree q
+
1, where the
degree of the tree is the common degree of all its vertices. To make things nontrivial
we shall assume that
q
0. (A homogeneous tree of degree one consists of two
vertices belonging to the same edge). With this hypothesis a homogeneous tree is
necessarily infinite. The case
q
= 1 corresponds to a double sequence of vertices
{xn :
n
E
Z}, such that the sets
{xn,Xn+d
are exactly the edges. The case
q
= 2
actually looks more like a tree, an appearance which is shared by homogeneous
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