# Connect the Dots Graph Theory

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```					Math 110: Class 1
Connect the Dots: Graph Theory
There’s an entire branch of mathematics called graph theory. Mathematicians interested in this subject study graphs. Note: One of the important aspects of mathematics is that each new concept should be carefully deﬁned. So what is a graph? Without being too technical we can deﬁne a graph as follows. Deﬁnition. A graph is a ﬁnite set of of vertices (or points) and edges (segments or arcs) connecting pairs of vertices. A graph is connected if we can travel from one vertex to any other along edges of the graph.

Z For simplicity we will, (1) assume that all graphs we discuss today are connected and (2) edges do not
cross each other, except where there is a vertex. Examples. Here are a few examples of connected graphs.
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Graphs have a wide variety of applications. A more complicated graph might represent any type of transportation network (airline, rail, road), or the relationships among people (‘friendship’ on Facebook), or even nodes on the internet. There are many interesting properties of graphs; today we focus on just one. Exercise 1. Draw three more connected graphs below. Try to make them look very diﬀerent. Ask your partner to verify that each is a graph.

Exercise 2. Draw an additional ﬁgure using vertices and edges which is not a connected graph. Carefully explain (write it out) why it is not. Does your explanation convince your partner?

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Exercise 3. Besides vertices and edges, each graph divides the plane into a number of disjoint ‘areas’ called regions (or faces). I have labeled the regions for the ﬁrst two ﬁgures below. Notice that in each ﬁgure there is one unbounded region or face that encompasses the remainder of the plane not already enclosed by the graph. For consistency, always label the unbounded region ﬁrst so you don’t forget it. Label the regions in the ﬁnal two ﬁgures. Then label the regions in your three graphs in Exercise 1. Finally, label the regions in your non-graph in Exercise 2.
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Exercise 4. Fill in the table listing the number of vertices v, edges e, and faces f for the four graphs above, your three graphs in Exercise 1 and those of your partner, and for the non-graphs of Exercise 2. (Check each other’s work.) In the ﬁnal column determine the Euler1 characteristic of the graph. The Euler characteristic of a graph G is the alternating sum χ(G) = v − e + f. Vertices v Graph A Graph B Graph C Graph D Graph 1 Graph 2 Graph 3 Partner 1 Partner 2 Partner 3 Non-Graph Partner’s Non-Graph Exercise 5. Are there any patterns in your table? Summarize your results in a clear statement. Edges e Faces f χ(G) = v − e + f

Exercise 6.

What should the next question be?

1 This was discovered the mathematician Leonhard Euler, 1707–1783. His name is pronounced as ‘oiler’. The symbol for the so-called Euler characteristic is the Greek letter χ (‘chi’).

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