Discrete Mathematics with Graph Theory (3rd Edition) 301

Discrete Mathematics with Graph Theory (3rd Edition) 301 -...

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Section 11.2 299 3. [BB] Every arc comes out of one vertex and goes into another, hence, adds one to the sum of all indegrees and one to the sum of all outdegrees. 4. The digraph is strongly connected-you can get from any vertex to any other along the circuit uvxwu-but it is not Eulerian because, for example, indeg u = 2 =I- 1 = outdeg u. 5. [BB] The answer is yes. Since 9 is an Eulerian graph, there exists an Eulerian circuit. Now just orient the edges of this circuit in the direction of a walk along it. 6. Since there are no cycles, any walk is a path, so it contains at most n vertices and n - 1 arcs. 7. The sum of the entries in row i of an adjacency matrix is the outdegree of vertex i; the sum of the entries in column i is the indegree of vertex i. 8 ~B](a) A ~ U ~ ~ ~ 1 (b) The (3,3) entry of A2 is 2 because there are two directed walks of length 2 from vertex 3 to vertex 3; namely, 323 and 343. The (1,4) entry of A2 is 1 because the only directed walk of length 2 from vertex 1 to vertex 4 is 134. (c) The (4, 2) entry of
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