Graph Properties - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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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?

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.