t03_y2008 - 7 Use Kruskal’s algorithm to ²nd all least...

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The University of Sydney MATH2969/2069 Graph Theory Tutorial 3 (Week 10) 2008 1. Draw all the spanning trees of this graph: 2. If A is the adjacency matrix of a simple graph G with vertex set { v 1 ,v 2 ,...,v n } , and ( A k ) ij is the ( i,j ) term of A k , show that ( A 2 ) ii = deg( v i ). 3. v 1 v 2 v 3 v 4 v 5 ( i ) Write down the adjacency matrix, A , for this graph. ( ii ) Calculate A 2 , and verify that the result proved in Question 2 holds. ( iii ) Find the number of di±erent walks of length 4 from v 5 to v 5 . ( iv ) Verify that the trace of A 3 is 6 times the number of triangles in the graph. 4. A graph G has adjacency matrix A = 0 1 1 2 0 1 0 0 0 1 1 0 0 1 1 2 0 1 0 0 0 1 1 0 0 . ( i ) Is G a simple graph? ( ii ) What is the degree sequence of G ? ( iii ) How many edges does G have? 5. Let A be the adjacency matrix of a bipartite graph. Prove that the diagonal entries of A 2 n +1 are all equal to 0, for any natural number n . 6. Find the number of spanning trees in each of the following graphs: ( i ) ( ii ) K 7 ( iii ) K 3 , 3 ( iv
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Unformatted text preview: ) 7. Use Kruskal’s algorithm to ²nd all least weight spanning trees for this 4 1 5 3 5 3 4 2 6 weighted graph. 8. Find a minimum cost spanning tree for the graph with this cost matrix. How many such trees are there? A B C D E F G H A 12 14 11 17 8 B 12 9 12 15 10 9 C 9 18 14 31 9 D 14 18 6 23 14 E 11 12 14 15 16 F 15 31 6 15 8 16 G 17 10 23 16 8 22 H 8 9 9 14 16 22 Extra questions 9. Find a minimum weight spanning tree in each of the following weighted graphs: ( i ) 3 4 1 2 6 6 5 7 ( ii ) 7 6 7 9 9 5 6 9 7 8 10. Use the Matrix Tree Theorem to verify the fact that the number of spanning trees of the complete graph K n is equal to the number of labelled trees on n vertices....
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t03_y2008 - 7 Use Kruskal’s algorithm to ²nd all least...

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