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 | cfwc | g fgc | wfg | wc wc | fgw | fgcg | fwc c | fwg| fwgc