326 Solutions to Exercises but it also must be cheaper than any other selection of edges connecting the six other towns to A. This is in fact the case, because ADEFCBG consists of the six edges of smallest weight in the entire graph! A 10. (a) The minimum cycle is ACBDA (or ADBCA, in reverse) with a total of90. C D c c DB c B A A A AA A (b) The minimum path is ADCBEA which, in reverse, is the same as AEBCDA, with a total of 220. A AAAAAAAAAAAAAAAAAAAAAAAA 11. (a) Give the tree n vertices. Then there are n -1 edges, so the sum of the degrees is 2(n -1). This gives 1 1 100 + 20(6) + - 120)4 + - 120)2 = -1) = 2n -2 so n = 138. The number of vertices of degree 2 is ~ (n - 120) = 9. (b) Let there be k vertices of degree 1 and n in all. Since the sum of the degrees is 21 £ 1
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