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practiceGraphs(1)

# practiceGraphs(1) - cycle 4 Show that a vertex c in the...

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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 the graph. 3. Show that a bipartite graph with an odd number of vertices does not have a Hamiltonian
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Unformatted text preview: cycle. 4. Show that a vertex c in the connected simple graph G is a cut 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. 5. Prove that any tree with 2 or more nodes must have at least two leaves. Note: A leaf vertex is a node in a tree with degree 1. 6. Show that if G is a bipartite graph with v vertices and e edges, then e ≤ v 2 4 ....
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