Trees
Lecture 3 : The most
beautiful graphs
Connected Graphs
Definition 3.1
A graph G is a connected graph if its
diameter diam(G) < +.
 Definition 3.2
A connected graph T is a tree if T
has no cycles as its subgraphs.

Bridges
A component of a graph is a maximal
connected subgraph.
 Let c(G) denote the number of
components in G. A graph is
connected if and only if c(G) = 1.
 An edge e of a graph G is a bridge if
c(G – e) > c(G).

Equivalent Statements

(1)
(2)
(3)
(4)
(5)
Theorem 3.3
Let T be a graph of order p. Then the
following statements are equivalent.
T is a tree.
T is connected and T has p – 1 edges.
T has no cycles and T has p – 1 edges.
T is connected and every edge of T is a bridge.
For each pair of vertices u and v in T there
exists a unique path from u to v.
Degree Sequences


Definition 3.4
Let G be a graph with vertex set
{v1,v2, …,vp}. {degG(vi) = di}i = 1 to p is called
a degree sequence of G. For
convenience, we may assume that d1  d2
 …  dp.
Theorem 3.5
If D = < d1,d2, …,dp> is a sequence of
non-decreasing positive integers such that
 di = 2p – 2, then there exists a tree T of
order p whose degree sequence is D.
Spanning Trees
Definition 3.6
A spanning subgraph H of a connected
G is a spanning tree if H itself a tree.
 Proposition 3.7
Every connected graph contains a spanning
tree.
 Proposition 3.8
Let T be a spanning tree of G and e is an edge
of T. Then there exists an edge f of G – T such
that T – e + f is a spanning tree of G.

The Number of
Spanning Trees

Theorem 3.9
The number of distinct spanning trees in the
complete graph of order p is equal to pp-2.
Proof. Let the complete graph of order Kp be defined
on [p] = {1,2,3, …,p}. Then for each spanning tree T
of Kp corresponds to a unique sequence <t1,t2, …,tp2> where ti is in [p] and the corresponding rule is as
follows:
Let s1 be the smallest integer which is a vertex of
degree one vertices in T and s1 is adjacent to t1 in T.
Then delete s1 from T and continue the above step to
find s2 and then t2 …. Now, the proof follows. (?)