# gr-t2 - NAME TEST2/MAD3305 Page 1 of 4 General directions...

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NAME: TEST2/MAD3305 Page 1 of 4 _________________________________________________________________ General directions: Read each problem carefully and do exactly what is requested. Show all your work neatly. Use complete sentences and use notation correctly. Make your arguments and proofs as complete as possible. Remember that what is illegible or incomprehensible is worthless. _________________________________________________________________ 1. (25 pts.) (a) What is a bridge?? [Yes, a definition is required.] (b) Give an example of a graph of order n and size n - 1 that is not a tree. (c) A certain tree of order n has only vertices of degree 3 and degree 1. How many degree 3 vertices does the tree have? (d) If G is a nonseparable graph with order at least 3, what is the best estimate that you provide for δ (G)? Why?? (e) Let T be a tree of order n 2 with k = (G). Let n i be the number of vertices of degree i for 1 i k. Then and . k i 1 n i n 2( n 1) k i 1 in i Show how to obtain a formula for the number of end-vertices of T.

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TEST2/MAD3305 Page 2 of 4 _________________________________________________________________ 2. (15 pts.) For the graph G below, determine the cut-vertices, bridges, and blocks of G. List the cut-vertices and bridges in
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