# So diamg max duv uv ε vg 3 5 pts list the r regular

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is denoted diam(G). So diam(G) = max { d(u,v): u,v ε V(G) }. _________________________________________________________________ 3. (5 pts.) List the r-regular graphs of order 5 for all possible values of r. They are all old friends. The r-regular graphs of order 5 are the empty graph of order 5 with r = 0, the 5-cycle with r = 2, and the complete K 5 C 5 graph with r = 4. [I’ll accept sketches, but I really K 5 wanted the "names".]

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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 s 1 : 4,3,3,3,2,1,0 s 2 : 2,2,2,1,1,0 s 3 : 1,1,1,1,0 [Graphic] _________________________________________________________________ 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
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.] Since δ (C 4 ) = 2 and (C 4 ) = 2, we need at least one vertex. There is not a 3-regular graph of order 5. So 6 vertices will be the best that we can do. _________________________________________________________________ 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. // Review?? (a): Theorem 2.14, page 51.
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