is denoted diam(G). So diam(G) = max { d(u,v): u,v
ε
V(G) }.
_________________________________________________________________
3. (5 pts.)
List the rregular graphs of order 5 for all
possible values of r. They are all old friends.
The rregular graphs of order 5 are the empty graph of order 5
with r = 0, the 5cycle
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
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_________________________________________________________________
4. (10 pts.)
Use the HavelHakimi 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 3regular 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 3regular 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 3regular
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 nontrivial 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 nonadjacent vertices u and v, then G is connected.
// Review??
(a): Theorem 2.14, page 51.
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 Summer '12
 Rittered
 Graph Theory, Vertex, subgraph

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