5.2 Trees

A tree is a connected graph without any cycles.

Theorem 5.2.1 A graph with n vertices is a tree if and only if it has n −1 edges and no cycles.

Proof

1. Suppose first that the graph is a tree2. removing an edge from a tree results in a graph having

two components, each of which is without a cycle.

3. after we have removed n−1 edges, what remains is a graph with n components; that is, one without any more edges.

1. suppose we have a set of n−1 edges2. Starting with a graph consisting of n components that is,

consisting of the n vertices and no edges3. Add these n-1 edges one at a time4. Since each added edge must be between vertices in

different components (for otherwise it would result in a cycle), it follows that each added edge decreases the number of components by one.

5. Thus, after the (n−1)th edge has been added, the graph has only one component and no cycles; in other words, it is a tree.

Proposition 5.2.1 Every tree has at least two leaves.

1. Let D be the sum of the degrees of all the vertices of a graph. And D = 2(n−1) when the graph is a tree

2. Suppose tree has only one leaveleave(degree=1)◎nonleaf degree is at least 2(degree sum=2*(n-1) )◎

3. The sum of the degrees is 1+2*(n-1) ≠2(n-1)4. A contradiction

Proof

Sum of the degrees=2x(5-1)=8

Lemma 5.2.1

Proof

Corollary 2.7.2 implies that

which is equivalent to the identity stated in the lemma.

Proposition 5.2.2 (Cayley’s Theorem) There are trees on a vertex set of size n.

Proof

N(B) denote the numberof elements of Bt(n) denote the number of trees on a set of n verticesLet Li denote the set of trees on V for which vertex i is a leaf, i = 1, . . . , n.

1. is true when n = 1 and n = 22. assume that it is true whenever the vertex set is of size smaller

than n3. Now consider the vertex set V = {1, . . . , n}, where n > 2

Since every tree has at least one leaf, we obtain from the inclusion–exclusion rule that

where the final equality follows from Lemma 5.2.1.