MATH 455 EXAM I
This exam is worth 100 points, with each problem worth 20 points. There are
problems on both sides of the page. Please complete Problem 1 and then any four
of the remaining problems. Unless indicated, you must justify your answer to receive
credit for a solution.
When submitting your exam, please indicate which problems you want graded by
writing them in the upper right corner on the cover of your exam booklet. You must
select exactly four problems; any unselected problems will not be graded, and if you
select more than four only the first four (in numerical order) will be graded.
(1) Please give the definitions requested. You may assume that we know the
definition of graph.
(a) (5 pts) Define walk and path.
(b) (5 pts) Define the distance function between vertices of a graph, and the
eccentricity of a vertex.
(c) (5 pts) Define tree and forest.
(d) (5 pts) Define bridge.
(2) Let H be the graph in Figure 1.
(a) (8 pts) Write the adjacency matrix for H.
(b) Compute the number of 3-step walks from
(i) (6 pts) Vertex 2 to itself.
(ii) (6 pts) Vertex 1 to vertex 4.
(3) (a) (12 pts) For each of the following Prüfer sequences, draw the corresponding labelled trees:
(i) 1, 2, 1
(ii) 1, 2, 3, 2, 1
(iii) 1, 2, 3, 4, 3, 2, 1
(b) (8 pts) Draw the tree for the Prüfer sequence 1, 2, 3, . . . , k −1, k, k −1, k −
2, . . . , 1.
(4) Let G be the graph built by starting with the complete graph K5 with the
vertices labelled 1, . . . , 5, and then deleting the two edges {1, 2} and {1, 3}.
(a) (2 pts) Draw a picture of G. Be sure to label the vertices.
(b) (4 pts) Write the adjacency matrix for G
(c) (6 pts) Write a matrix M such that the absolute value of the determinant
of M counts the spanning trees in G.
(d) (8 pts) Compute the number of spanning trees in G. You can use your
result from (c), or you can do it a different way if you like.
Date: Thursday, 23 February 2012.
(5) (20 pts) Show that if v is a vertex of a tree T , then in the Prüfer sequence for
T the label of v appears deg(v) − 1 times. (Hint: There are two cases: (a)
v survives to be a vertex of the final K2 . (b) v is deleted before reaching the
final K2 .)
(6) Let G be the complete graph K6 . Label the vertices 1, . . . , 6, and suppose
that the edge between vertex i and vertex j has the weight |i − j|.
(a) (4 pts) Draw G and indicate what the weights of the edges are.
(b) (8 pts) Find a minimal cost spanning tree in G.
(c) (8 pts) Is the tree from (b) unique? If not, find another one. If so, explain
why.
(7) (20 pts) Let G be the complete graph Kn . Suppose G′ is another graph with
G as a subgraph. Suppose that if all the edges in G are deleted, then G′
becomes a forest. How many spanning trees does G′ have, and why?
(8) Let G be the complete graph K7 .
(a) (10 pts) Find a partition of the edges of E into disjoint cycles. (Hint:
there is a nice one that has 3 cycles, each of length 7. But there are lots
of others.)
(b) (10 pts) Use your answer to (a) to construct an Eulerian circuit in G.
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3
4
Figure 1.