Chapter-7 - MATH 45 Contemporary Mathematics Lecture Notes...

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MATH 45: Contemporary Mathematics Lecture Notes The Mathematics of Networking Fresno City College Geoffery L. McDonald Spring 2011
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Contents The Network Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Spanning Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Kruskal’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 iii
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iv CONTENTS
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THE NETWORK PROBLEM 1 The Network Problem For the past few chapters, now, we have been working with graphs. We took a look at routing problems (using Euler circuits and traveling every edge of a graph once, starting and ending at the same vertex), and we have also taking a gander at the Traveling Salesman Problem (using Hamilton circuits and traveling every vertex of a graph once, starting and ending at the same vertex). Now, let’s take a look at Networking Problems. Can we find a way to link all points (vertices), so that we can go from any one vertex to any other vertex? This time; though, we may not necessarily have a circuit in our graph. Thus for this chapter, we need some new terminology Network: This is just another name for a connected graph . A network could have circuits, but it could also not have circuits. Weighted Network: A network that has weights associated with its edges. Tree: A network with no circuits. Trees are networks; though, so they must be connected graphs, but trees cannot have any circuits. Trees are very important, and have been extensively studied. They are a very special type of graph and have some unique properties:
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