Homework4

# Homework4 - Show there is a labeling of the vertices of G...

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HOMEWORK 4 8 PROBLEMS DUE: WEDNESDAY, MARCH 2, 2011 (1) Draw the tree whose Pr¨ufer code is (1 , 1 , 1 , 1 , 6 , 5). (2) Determine which trees have Pr¨ufer codes that have distinct values in all positions. (3) Let G be a connected graph which is not a tree and let C be a cycle in G . Prove that the complement of any spanning tree of G contains at least one edge of C . (4) Suppose a graph G is formed by taking two disjoint connected graphs G 1 and G 2 and identifying a vertex in G 1 with a vertex in G 2 . Show that τ ( G ) = τ ( G 1 ) τ ( G 2 ). (5) Assume the graph G has two components G 1 and G 2
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Unformatted text preview: . Show there is a labeling of the vertices of G such that the adjacency matrix of G has the form A ( G ) = ± A ( G 1 ) A ( G 2 ) ² . (6) An m-fold path , mP n , is formed from P n by replacing each edge with a multiple edge of multiplicity m . An m-fold cycle , mC n , is formed from C n by replacing each edge with a multiple edge of multiplicity m . (a) Find τ ( mP n ) (b) Find τ ( mC n ) (7) Find τ ( K 2 , 3 ). (8) Use the Matrix-Tree Formula to compute τ ( K 3 ,n )....
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