145HW5 - be read in one direction only, specied by an...

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Homework assignment Number 5 Math 145 // due on 02/09/2007 1. Problems Section 7.3: 5, 8, 9, 10, 12. 2. Can we connect 15 computers in such a way that each is connected with exactly 3 other computers? 3. For which values of n does the complete graph K n have an Eulerian walk? 4. Find a hamiltonian cycle in the graph formed by the edges and vertices of an ordinary cube. 5. In this problem, we define and study the simple properties of the adjacency matrix of a simple graph. Let G = ( V, E ) be a graph with V = { 1 , . . . , n } . We define the adjacency matrix M of size n × n by M ij = 1 if ( ij ) E, M ij = 0 otherwise. (a) Give the adjacency matrix of the graph G = ( V, E ) where V = { 1 , 2 , 3 , 4 } and E = { (13) , (14) , (23) , (24) } and that of the complete graph K n . (b) In questions (b), (c) and (d), we study the general case. Show that M is a symmetric matrix. (c) Show that M k ij = ± { walks of length k from i to j } . Hint: use induction on k. (d) Give an interpretation of the sum n j =1 M ij for each i = 1 , . . . , n. (e) From now on ( (e) and (f) ), we consider the case of an oriented graph: each edge can
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Unformatted text preview: be read in one direction only, specied by an arrow. For instance, consider the graph G = ( V, E ) where V = { 1 , 2 , 3 } and E = { (1 2) , (2 1) , (2 3) } . Then there exists a path from 1 to 3 but no path from 3 to 1. G is still a simple graph, as (1 2) = (2 1) . Its adjacency matrix is then M = 1 1 1 . In other words, we set M ij = 1 if ( i j ) E, and M ij = 0 otherwise. Note that in general M is no longer symmetric. Let then G = ( V, E ) be an oriented graph with V = { 1 , . . . , n } . Consider the adjacency matrix M of G . Assume that M n = 0 where 0 is here the matrix with all its entries being null. Show that G has then a cycle. Hint: prove that (c) still holds for an oriented graph. (f) Examine the reciprocal. 1...
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