Problem Set 9
Section 7.5
#10 Let R be the relation on the set of ordered pairs of positive integers such that
((a,b), (c,d))  R if and only if ad = bc. Show that R is an equivalence relation.
#18 Determine whether the relations represented by the following zero-one matrices are
equivalence relations:
a)
b)
c)
1010
1110
111 
0101
1110
011




 
1010
1110
111 




0101
0001
Section 8.1
#10 Simplify each undirected graph below:
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Section 8.2
#18: Draw the following graphs:
a) K7
b) K1,8
c) K4,4
d) C7
e) W7
f) Q4
#22: Determine whether the graph is bipartite:
a
b
f
c
e
d
#26 How many edges does a graph have if it has vertices of degree 4, 3, 3, 2, 2?
Section 8.3
#2 Use an adjacency list to represent the graph below:
#20 Find the adjacency matrix of the graph below:
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#24 Draw the graph represented by this adjacency matrix:
a
b
c
d
a
0
1
2
1
b
2
2
1
0
c
3
2
1
0
d
0
1
0
2
#36 Determine whether the pair of graphs below is isomorphic; exhibit an isomorphism or
provide a rigorous argument that none exists.
Section 8.4
#4 Determine whether the graph is connected
Section 8.5
#2 Determine whether the graph has an Euler circuit; construct such a circuit if it exists:
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#14 Determine whether the picture below can be drawn with a pencil in a continuous motion
without lifting the pencil or retracing part of the picture:
#30 Determine whether the graph has a Hamilton circuit. If it does, find such a circuit; if it
doesn't, give an argument to show why no such circuit exists:
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