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Graphs and Trees

Graphs and Trees. This handout: Terminology of Graphs Applications of Graphs. Derive properties, get applications. Using the tools of logical reasoning. Mathematical objects. numbers. sets. functions. graphs. The process of mathematical reasoning.

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Graphs and Trees

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  1. Graphs and Trees • This handout: • Terminology of Graphs • Applications of Graphs

  2. Derive properties, get applications Using the tools of logical reasoning Mathematical objects numbers sets functions graphs The process of mathematical reasoning • We considered the first three types of mathematical objects • Next: Graphs, their properties and applications

  3. Terminology of Graphs • A graph (or network) consists of • a set of points • a set of lines connecting certain pairs of the points. The points are called nodes (or vertices). The lines are called arcs (or edges or links). • Example:

  4. Graphs in our daily lives • Transportation • Telephone • Computer • Electrical (power) • Pipelines • Molecular structures in biochemistry

  5. a c b e f Terminology of Graphs • Each edge is associated with a set of two nodes, called its endpoints. Ex:a and b are the two endpoints of edge e • An edge is said to connect its endpoints. Ex: Edge e connects nodes a and b. • Two nodes that are connected by an edge are called adjacent. Ex: Nodes a and b are adjacent.

  6. Terminology of Graphs: Paths • A path between two nodes is a sequence of distinct nodes and edges connecting these nodes. Example: • Walksare paths that can repeat nodes and arcs. a b

  7. A little history: the Bridges of Koenigsberg • “Graph Theory” began in 1736 • Leonhard Eüler • Visited Koenigsberg • People wondered whether it is possible to take a walk, end up where you started from, and cross each bridge in Koenigsberg exactly once

  8. The Bridges of Koenigsberg A 1 2 3 B 4 C 5 6 7 D Is it possible to start in A, cross over each bridge exactly once, and end up back in A?

  9. The Bridges of Koenigsberg A 1 2 3 B 4 C 5 6 7 D Translation into a graph problem: Land masses are “nodes”.

  10. A B C D The Bridges of Koenigsberg 1 2 3 4 6 5 7 Translation into a graph problem: Bridges are “arcs.”

  11. A B C D The Bridges of Koenigsberg 1 2 3 4 6 5 7 Is there a “walk” starting at A and ending at A and passing through each arc exactly once? Such a walk is called an eulerian cycle.

  12. A B C D Adding two bridges creates such a walk 3 8 1 2 4 6 5 9 7 Here is the walk. A, 1, B, 5, D, 6, B, 4, C, 8, A, 3, C, 7, D, 9, B, 2, A Note: the number of arcs incident to B is twice the number of times that B appears on the walk.

  13. A 3 8 1 2 4 B 6 5 9 C 7 D 4 Existence of Eulerian Cycle The degree of a node is the number of incident arcs 6 4 4 Theorem. An undirected graph has an eulerian cycle if and only if (1) every node degree is even and (2) the graph is connected (that is, there is a path from each node to each other node).

  14. 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?

  15. Terminology of Graph: Paths • A path between two nodes is a sequence of distinct nodes and edges connecting these nodes. Example: • Two nodes are calledconnected 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

  16. 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?

  17. 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.

  18. An application of graphs in solving a puzzle fwgc | fwg | c fwc | g fgc | w fg | wc 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) c | fwg | fwgc wc | fg w | fgc g | fwc

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