EECS 310, Fall 2011
Instructor: Nicole Immorlica
Practice Problems: Graph Theory
1. Does there exist a (simple) graph with five vertices of the following degrees? If so, draw
such a graph. If not, explain why.
(a) 3
,
3
,
3
,
3
,
2
Solution:
Yes, draw a complete graph on 4 vertices, and then split one edge
by adding a vertex in the middle.
(b) 1
,
2
,
3
,
4
,
5
Solution:
No, the sum of degrees must be even, but 1 + 2 + 3 + 4 + 5 is odd.
(c) 0
,
1
,
2
,
2
,
3
Solution:
Yes, draw a complete graph on 3 vertices, add a vertex with an edge
to one of the first three, and another vertex with no edges.
(d) 0
,
1
,
1
,
2
,
4
Solution:
No, removing the degree zero vertex leaves a graph on
n
= 4 vertices
with max degree Δ(
G
) = 4, but in any simple graph, Δ(
G
)
≤
n

1.
2. Draw the graph with the following adjacency matrix:
.
v
1
v
2
v
3
v
4
v
5
v
1
0
1
0
1
0
v
2
1
0
1
1
0
v
3
0
1
0
0
1
v
4
1
1
0
0
1
v
5
0
0
1
1
0
Solution:
Draw a 5cycle where the order of vertices is
v
1
, v
2
, v
3
, v
5
, v
4
and add an
edge between
v
2
and
v
4
.
(a) What is the length of the shortest path between vertices
v
1
and
v
3
?
Solution:
Take powers of the adjacency matrix until the corresponding entry
is nonzero to get the answer: 2.
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(b) How many walks of length 3 are there between vertices
v
1
and
v
2
?
Solution:
Cube the adjacency matrix and read off the entry in the 1’st row
and 2’nd column to get the answer: 4.
(c) Derive a formula that relates the adjacency matrix to the number of triangles in
the graph.
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 Fall '11
 N/A
 Economics, Graph Theory

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