Graphs and Trees

This handout:

• Total degree of a graph

• Applications of Graphs

Graph properties• Definition: The total degree of a graph is the sum of the

degrees of all its nodes.

• Theorem: If G is any graph, then the total degree of G equals twice the number of edges of G:

the total degree of G = 2 (the number of edges of G)• Corollary 1: The total degree of a graph is even.• Corollary 2: In any graph there are an even number of

vertices of odd degree.

• Application to an Acquaintance Graph: Is it possible in a group of five people

for each to be friends with exactly three others?

Terminology of Graph: Paths

• A path between two nodes is a sequence of distinct nodes and edges connecting these nodes.Example:

• Two nodes are called connected if there is a path between them.

• Fact: For any two nodes a and b of a graph, there is an efficient way to determine whether a and b are connected or not.

a

b

An application of graphs in solving a puzzle

From an initial position on the left bank of a river,

a ferryman wants to transport

a wolf, a goat, and a cabbage to the right bank.

Ferryman’s boat is only big enough

to transport one object at a time, other than himself.

For obvious reasons,

• the wolf cannot be left alone with the goat;

• the goat cannot be left alone with the cabbage.How should the ferryman proceed?

An application of graphs in solving a puzzle

To solve the puzzle, create the following graph: Create a node for each allowable arrangement.

E.g., ( fg | wc ) is an allowable arrangement

since the ferryman and the goat are on the left bank,

and the wolf and the cabbage are on the right bank.

Create an edge between two nodes if it is possible to go from the arrangement of one node to the arrangement of the other node by a single ferry trip.E.g., there is an arc between nodes ( fgw | c ) and ( w | fgc ) because the transition from the first node to the second node can be realized

by a single trip of the ferryman with the goat

from the left bank to the right bank.

An application of graphs in solving a puzzle

The resulting graph is:

To transport everything from the left bank to the right bank, we need to find a path from node ( fwgc | ) to node ( | fwgc ) in the graph.

There are two this kind of paths. One of them:

(fwgc | ) (wc | fg) (fwc | g) (w | fgc) (fwg | c) (g | fwc) (fg | wc) ( | fwgc)

fwgc | fwg | c fwc | g fgc | w fg | wc

wc | fg w | fgc g | fwc c | fwg | fwgc