Sketch two graphs G and H that have the degree sequence
s: 2, 2, 2, 2, 2, 2 and have the same order and size, but are not
isomorphic.
Explain briefly how one can readily see that the
graphs are not isomorphic.
(c)
Explicitly realize C
5
and its complement below. [You may
provide carefully labelled sketches.]
Next, explicitly define an
isomorphism from C
5
to its complement that reveals that C
5
is
selfcomplementary.
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7. (10 pts.)
Find a minimum spanning tree for the weighted
graph below by using only Prim’s algorithm and starting with the
vertex g.
When you do this, list the edges in the order that you
select them from left to right.
What is the weight w(T) of your
minimum spanning tree T?
_________________________________________________________________
8. (15 pts.)
(a)
If G is a nontrivial graph, how is
κ
(G), the
vertex connectivity of G, defined?
(b)
If G is a nontrivial graph, it is not true generally that if
v is an arbitrary vertex of G, then either
κ
(
Gv
)=
κ
(
G
)1o
r
κ
(
Gv
)=
κ
(G).
Give a simple example of a connected graph G
illustrating this.
[A carefully labelled drawing with a brief
explanation will provide an appropriate answer.]
(c)
Despite the example above, if G is a nontrivial graph and v
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 Summer '12
 Rittered
 Graph Theory, Vertex, 10 pts, κ, 15 pts, 25 pts

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