E suppose that g is a graph of order 25 and size 99

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(e) Suppose that G is a graph of order 25 and size 99. From this information, we know that any path in G can be no longer than what number l? Provide the best upper bound.
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TEST1/MAD3305 Page 2 of 4 _________________________________________________________________ 2. (20 pts.) Provide mathematical definitions for each of the following terms. (a) A graph G: (b) Subgraph: (c) Spanning Subgraph: (d) Bipartite Graph: (e) Diameter: _________________________________________________________________ 3. (5 pts.) List the r-regular graphs of order 5 for all possible values of r. They are all old friends.
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TEST1/MAD3305 Page 3 of 4 _________________________________________________________________ 4. (10 pts.) Use the Havel-Hakimi Theorem to construct a graph with degree sequence s: 7,5,4,4,4,3,2,1 _________________________________________________________________ 5. (10 pts.) Use the ideas from the proof of Theorem 2.7, to construct a 3-regular graph G that contains K 3 as an induced subgraph. Show each stage of the construction. _________________________________________________________________ 6. (5 pts.) Sketch a graph G that has the following adjacency matrix: A G 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0
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TEST1/MAD3305 Page 4 of 4 _________________________________________________________________ 7. (5 pts.) Construct a 3-regular graph G of minimum order that contains C 4 as an induced subgraph. [Use the ideas of Paul Erdos and Paul J. Kelly.] _________________________________________________________________ 8. (10 pts.) Prove exactly one of the following propositions. Indicate clearly which you are demonstrating. (a) If G is a non-trivial graph, then there are distinct vertices u and v in G with deg(u) = deg(v). (b) If G is a graph of order n and deg(u) + deg(v) n - 1 for each pair of non-adjacent vertices u and v, then G is connected. _________________________________________________________________ 9. (10 pts.) (a) Suppose G is a bipartite graph of order at least 5. Prove that the complement of G is not bipartite. [Hint: At least one partite set has three elements. Connect the dots?] (b) Display a bipartite graph G of order 4 and its bipartite complement. Label each appropriately and give partite sets for each bipartite graph.
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  • Summer '12
  • Rittered
  • Graph Theory, Vertex, pts, Paul Erdős

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