Chapter 7 The Mathematics of Networks

Chapter 7 The Mathematics of Networks - Excursions in...

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1 Excursions in Modern Mathematics Sixth Edition Peter Tannenbaum

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2 Chapter 7 The Mathematics of Networks The Cost of Being Connected
3 The Mathematics of Networks Outline/learning Objectives To identify and use a graph to model minimum network problems. To classify which graphs are trees. To implement Kruskal’s algorithm to find a minimal spanning tree.

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4 The Mathematics of Networks Outline/learning Objectives To understand Torricelli’s construction for finding a Steiner point. To recognize when the shortest network connecting three points uses a Steiner point. To understand basic properties of the shortest network connecting a set of (more than three) points.
5 The Mathematics of Networks 7.1 Trees

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6 The Mathematics of Networks Network Network Another name for a connected graph. Tree Tree A network with no circuits. Spanning Tree Spanning Tree A subgraph that connects all the vertices of the network and has no circuits. Minimum Spanning Tree (MST) Minimum Spanning Tree (MST) Among all spanning trees of a weighted network , one with the least total weight.
7 The Mathematics of Networks Tree or Not? The graphs in (a) and (b) are disconnected, so they are not even networks, let alone trees.

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8 The Mathematics of Networks Tree or Not? The graphs in (c) and (d) are networks that have circuits so neither of the is a tree.
9 The Mathematics of Networks Tree or Not? The graphs in (e) and (f) are networks with no circuits, so they are indeed trees.

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10 The Mathematics of Networks Tree or Not? The structure of a family tree (g) and the structure formed by the bonds of some molecules (h) are also trees.
11 The Mathematics of Networks Summary of Key Properties Property 1 In a tree, there is one and only one path joining any two vertices. If there is one and only one path joining any two vertices of a graph, then the graph must be a tree.

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12 The Mathematics of Networks Summary of Key Properties Property 2 In a tree, every edge is a bridge. If every edge of a graph is a bridge, then the graph must be a tree.
13 The Mathematics of Networks Summary of Key Properties Property 3 A tree with N vertices has N – 1 edges. If a network has N vertices and N – 1 edges, then it must be a tree.

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14 The Mathematics of Networks Notice that a disconnected graph (not a network) can have N vertices and N – 1 edges.
15 The Mathematics of Networks 7.2 Spanning Trees

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16 The Mathematics of Networks Summary of Key Properties Property 4 If a network has N vertices and M edges, then M N 1. [ R = M – ( N – 1) as the redundancy of the network.] If M = N – 1, the network is a tree; if M N – 1, the network has circuits and is not a tree. (In other words, a tree is a network with zero redundancy and a network with positive redundancy is not a tree.
The Mathematics of Networks Counting Spanning Trees The network in (a) has N = 8 vertices and M = 8 edges. The redundancy of the network is

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Chapter 7 The Mathematics of Networks - Excursions in...

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