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

• Notes
• 4

This preview shows pages 1–4. Sign up to view the full content.

(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.

This preview has intentionally blurred sections. Sign up to view the full version.

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.
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

This preview has intentionally blurred sections. Sign up to view the full version.

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.
This is the end of the preview. Sign up to access the rest of the document.
• Summer '12
• Rittered
• Graph Theory, Vertex, pts, Paul Erdős

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern