# Ch6 - MGF 1107 Lecture Outline Unit IV Routes and Networks...

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MGF 1107 – Lecture Outline Unit IV – Routes and Networks; Growth and Decay (Chapters 6 and 12) Chapter 6 – Routes and Networks These Notes are taken from A Mathematical View of our World , Parks, Musser, et. al., 1 st custom ed., Thomson Publishing and the accompanying Instructor’s Resource CD. All rights reserved by the publisher. Sec 6.1 – Routing Problems Goals o Study graphs, vertices, and edges o Study paths and circuits o Study connected graphs o Use Euler’s Theorem Graphs o A ___________ is a collection of one or more points, called _____________, and the paths connecting them, called ________. o The singular of vertices is vertex. o The vertices are drawn as _________. o The edges are drawn as straight or curved __________. o Each edge has two ends with a vertex at each end, or else it is a __________ connecting a vertex to itself. o Two graphs are considered the same (___________________) if they have the same number of vertices connected in the same way, even if they look different. See figure 6.4, p. 342. o Two edges may appear to cross, but they do not actually intersect unless there is a ___________. o Graphs can be drawn to represent real-life connections, such as roads between cities. 1

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The following chart shows travel times between various places. Library Post office School Home Library - 12 9 18 Post office 12 - 10 20 School 9 10 - 14 Home 18 20 14 - Draw a graph to illustrate this situation. o Two vertices are called _____________________________ if there is an edge connecting them. o The ______________ of a vertex is the total number of edges at the vertex. o Find the degree of a vertex by counting the number of edges leaving or arriving at the vertex. o A _________ contributes ________ to the degree of a vertex. (A _______ counts as a single edge, however). o If d is the sum of all the degrees of the vertices in a graph and e is the number of edges in the graph, then ______________ . o Note: This theorem is true because each edge adds 2 to the degree total. Work Homework #6 (pg 353) 2
Paths and Circuits o A ___________ is a route in a graph. o Each vertex used is adjacent to the next vertex used. o Each edge in the graph is used _______________ one time. o A path is written as a list of vertices. o A _____________ is a path that begins and ends at the same vertex. o A path that uses every edge of a graph exactly once is called an ______________________. o Note: Euler is pronounced “oiler.” o A circuit that uses every edge of a graph exactly once is called an . Work Homework #18 (pg 355) 3

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Connected Graphs o A graph is said to be if, for every pair of vertices, there is a path that contains them. o In a connected graph, we can travel from any vertex to any other vertex. o
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## This note was uploaded on 11/07/2011 for the course MGF 1107 taught by Professor Wooland during the Summer '08 term at FSU.

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Ch6 - MGF 1107 Lecture Outline Unit IV Routes and Networks...

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