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Exploring Directed Graphs and Adjacency Matrices in Finite Graphs

A finite graph is comprised of nodes connected by edges, represented using an adjacency matrix. In this matrix, a '1' indicates a direct connection between nodes. Directed graphs, which include information about the direction of edges, provide useful applications, like analyzing flight routes or communication networks. Calculators can speed up matrix computations, enabling us to explore complex scenarios such as determining communication paths or travel options efficiently. Homework tasks are integrated for deeper understanding of these concepts.

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Exploring Directed Graphs and Adjacency Matrices in Finite Graphs

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  1. 12.3 Directed Graphs

  2. A finitegraph is a set of points, called nodes, connected by a set of lines, called edges. We can represent the graph in an adjacencymatrix. If the nodes are connected, you place a 1 in the matrix. Ex: • TO: • n1n2n3n4 • n2 • n1 • n3 FROM: • n4 A finite graph may need to include info about direction. We get a directedgraph. *Flights work this way. From Long Beach to Orlando on Jet Blue 

  3. Ex 1) Create the adjacency matrix. • TO: • C N S Chicago New York FROM: St. Louis What options do you have if you are making 2 flights? You would have the same matrix, but there would be two of them.

  4. If we continued this, it would be a lot of multiplying. We can use our graphing calculators to speed things up! 2ndx–1 is MATRIX  EDIT choose 1: [A] enter dimensions 3 × 3 Now enter each element. When done, go 2nd MODE to QUIT Calculate [A]2 MATRIX  NAMES  1: [A] matches! 

  5. Ex 2) Suppose the nodes in the diagram represent people and the directed edges mean the first person knows the second person’s phone number. a) Interpret the diagram in a matrix. Betty Charles Al • 2nd person: • A B C D E F David Fred 1st person: Ellen

  6. Ex 2) cont… In how many ways can a message get from Betty to Ellen in 3 or fewer calls? (*Use your calculator!) • A B C D E F • A B C D E F • A B C D E F No ways 2 ways 1 way so 3 different ways

  7. Ex 2) cont… c) Is it possible to get a message from Fred to Charles? If yes, what is the minimum number of calls? • A B C D E F • A B C D E F 1 call No way 2 calls 1 way Yes, 2 calls minimum

  8. Homework #1203 Pg 616 #1–5, 7, 9, 11, 15, 17–20, 22–32

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