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a4-17-8

# a4-17-8 - Section 48 1 Let G be the graph in the gure(a How...

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Section 48 1. Let G be the graph in the °gure. (a) How many di/erent paths are there from a to b ? Each path has to go through that point w in the middle (the waist of the graph). There are 5 paths from a to w and 3 paths from w to b , so there are 15 = 3 ° 5 paths from a to b . (b) How many di/erent walks are there from a to b . In general, if there is any walk between two vertices, there are in°nitely many, because you can just walk back and forth. So we could go from a to w to b , back to w , back to b back to w , back to b , and so on. 4. Let n ± 2 be an integer. Form a graph G n whose vertices are all the two element subsets of f 1 ; 2 ; : : : ; n g . In this graph we have an edge between distinct vertices f a; b g and f c; d g exactly when f a; b g \ f c; d g = ; . (a) How many vertices does G n have ? A vertex is a two element subset of f 1 ; 2 ; : : : ; n g so there are ° n 2 ± = n ( n ² 1) = 2 of them. (b) How many edges does G n have ? We have to count the number of ways we can choose two disjoint vertices. We can pick the °rst vertex in any of ° n 2 ± ways. Hav- ing picked the °rst vertex, we can pick the second one in any of ° n ² 2 2 ± ways (we can±t reuse any of the number we used in the °rst vertex because the two vertices must be disjoint). So the number ° n 2 ±° n ² 2 2 ± counts how many ordered pairs of disjoint vertices we can form. We have to divide this by 2 because the edges correspond to an unordered pairs of disjoint vertices. So the answer is ° n 2 ±° n ² 2 2 ± 2 1

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Notice that this number is zero for n = 2 and n = 3 . For n = 4 it is 6 ° 1 = 2 = 3 and for n = 5 it is 10 ° 3 = 2 = 15 .
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a4-17-8 - Section 48 1 Let G be the graph in the gure(a How...

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