practiceGraphs

# practiceGraphs - the graph 3 Show that a bipartite graph...

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EECS 310, Fall 2011 Instructor: Nicole Immorlica Practice Problems: Graph Theory 1. Does there exist a (simple) graph with ﬁve vertices of the following degrees? If so, draw such a graph. If not, explain why. (a) 3 , 3 , 3 , 3 , 2 (b) 1 , 2 , 3 , 4 , 5 (c) 0 , 1 , 2 , 2 , 3 (d) 0 , 1 , 1 , 2 , 4 2. Draw the graph with the following adjacency matrix: . v 1 v 2 v 3 v 4 v 5 v 1 0 1 0 1 0 v 2 1 0 1 1 0 v 3 0 1 0 0 1 v 4 1 1 0 0 1 v 5 0 0 1 1 0 (a) What is the length of the shortest path between vertices v 1 and v 3 ? (b) How many walks of length 3 are there between vertices v 1 and v 2 ? (c) Derive a formula that relates the adjacency matrix to the number of triangles in
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Unformatted text preview: the graph. 3. Show that a bipartite graph with an odd number of vertices does not have a Hamiltonian cycle. 4. Show that a vertex c in the connected simple graph G is a cute vertex if and only if there are vertices u and v , both diﬀerent from c , such that every path between u and v passes through c . Note: A cut vertex is a vertex v in G such that the removal of v from G results in disconnecting the graph....
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## This note was uploaded on 01/15/2012 for the course ECON 101 taught by Professor N/a during the Fall '11 term at Middlesex CC.

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