[The sets U and W are called partite sets.]
(e)
Diameter:
The greatest distance between any two
vertices of a connected graph G is called the diameter of G and
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
w
i
t
hr=0
,t
h
e 5cycle
withr=2
,a
n
dt
h
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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
010110
101001
010100
101010
100101
010010
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
)=2a
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 Summer '12
 Rittered
 Graph Theory, Vertex, subgraph

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