Question

Question 4t Write the correct integer values in the boxes. For this question, working is not required and will not be marked. This question is about the number of spanning trees of a graph. In a lecture we used complementary counting to calculate that the graph depicted at left has exactly eight spanning trees. By adding just one more edge to this graph we arrive at the complete graph K depicted at right. A spanning tree has -1 3 edges and so we first count the total number of sets of three edges; our extra edge takes this from ( to () Now we count the number of sets to discard because they do not create a tree. The extra edge means there are a couple more of these. This leads to: (a) The number of spanning trees of K4 is Moving up to 5 vertices, we next consider the graph G depicted at right, which has five vertices and six edges. Again using complementary counting we need to find two numbers: number, not formula (b) The total number of sets of edges we need to consider is (c) The number of these sets that do not provide a tree is By subtracting answer (c) from answer (b) you will get the number of spanning trees for G. You can check your answers by drawing and counting all these trees. Finally consider the case of the complete graph Ks depicted at right. It again has five vertices, but now ten edges, so there wil be a lot more spanning trees. Counting as before: number, not formula (d) The total number of sets of edges we need to consider is (e) The number of these sets which need to be discarded because they form disconnected subgraphs (and hence cannot be trees) is (f) The number of these sets which need to be discarded because they form connected but not circuit-free subgraphs (and hence cannot be trees) is Hint: Two types of circuits are involved. There are a lot. By subtracting answers (e) and (f) from answer (d) you wil get the number of spanning trees for K. You might spot a pattern by comparing the corresponding values for K4 and K3. The simple formula that applies, and especially the proof of that formula, is one of the jewels of graph theory. Try Googling it

Answer 1

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