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Homework 12 Solutions W11

# Homework 12 Solutions W11 - EECS 203 Homework 12 Solutions...

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EECS 203: Homework 12 Solutions Section 9.3 1. (E) 6,8 (6) a b c d e a 0 1 0 1 0 b 1 0 0 1 1 c 0 0 0 1 1 d 1 1 1 0 0 e 0 1 1 0 0 (8) a b c d e a 0 1 0 1 0 b 1 0 1 1 1 c 0 1 1 0 0 d 1 0 0 0 1 e 0 0 1 0 1 2. (E) 14 a b c d a 0 3 0 1 b 3 0 1 0 c 0 1 0 3 d 1 0 3 0 3. (E) 36,38 (36) The graphs are not isomorphic because the right graph’s v 2 has a degree of 4 whereas none of the left graph’s nodes have any nodes with a degree of 4. (38) The graphs are isomorphic. An example of corresponding vertices are { v 2 u 3 } , { v 3 u 4 } , { v 5 u 2 } , { v 1 u 1 } and { v 4 u 5 } . 4. (C) 52 In order for graph G to be self-complementary, the number of edges in G and G must be equal. Because the total number of distinct connections in both G and G is n ( n - 1) 2 , meaning that there can be n ( n - 1) 4 edges in each of G and G . It means that either n or n - 1 must be divisible by 4 in order for this quantity to be an integer. Therefore, if G is self-complementary, n is either 0 or 1 modulo 4 . 5. (M) 66 Let graphs G and H to be isomorphic graphs. If G is a bipartite graph, then the nodes in G can be divided into two groups with no edges between any pair of nodes that belong to the same group. Because of the mapping (due to isomorphism), one can replace nodes in G with equivalent ones from H . Therefore, the new nodes from H must have the same relationship with other nodes in the same way as the original nodes from G , which means H must also be a bipartite graph.

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Section 9.4 6. (E) 12bc (12b) The graph is strongly connected because there are paths from one node to any other nodes. (Look at the cycle a b c d e f a ). (12c) The graph is neither strongly nor weakly connected because the vertices { a, g, c, d } and vertices { b, e, f } are completely disconnected. 7. (E) 20 The graphs G and H are isomorphic. Possible steps to find corresponding vertices with paths are as followed: (a) There are two cycles with length of four and two cycles with length of three in H : { v 1 , v 3 , v 4 , v 8 } , { v 8 , v 4 , v 5 , v 7 } , { v 2 , v 1 , v 3 } and { v 5 , v 7 , v 6 } . (b) There are two cycles with length of four and two cycles with length of three in G : { u 2 , u 3 , u 4 , u 5 } , { u 7 , u 4 , u 5 , u 6 } , { u 1 , u 2 , u 3 } and { u 6 , u 7 , u 8 } . (c) Note that the two cycles with length of four share two nodes. Therefore, { v 4 , v 8 } are associated with { u 4 , u 5 } .
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