Pick such a vertex, say w,
and let T = (V(G), E(T
0
)
∪
{vw}).
Then T is a spanning tree for G. Since G was an
arbitrary connected graph of order k+1, S(k+1) follows.
Thus, S(k)
⇒
S(k+1).
Since k was arbitrary, we have (
∀
k
≥
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_________________________________________________________________
5. (10 pts.)
Apply Kruskal’s algorithm to find a minimum
spanning tree in the weighted graph below.
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?
There are several
different correct
solutions.
First, in
some order you will
take two of the three
edges from the "1" 3
cycle:
de, ef, fd.
Then you will need
both "2" edges in some
order: ae and ac.
Then exactly one
of the two "3" edges:
ab or bd.
For any of the 6
possible trees T,
w(T) = 9.
_________________________________________________________________
6. (15 pts.)
(a)
Suppose G
1
and G
2
are nontrivial graphs.
What does it mean mathematically to say that G
1
and G
2
are
isomorphic?? [This is really a request for the definition!]
Two graphs G
1
and G
2
are isomorphic if there is a bijection
φ
: V(G
1
)
→
V(G
2
) such that uv
ε
E(G
1
) if, and only if
φ
(u)
φ
(v)
ε
E(G
2
).
(b)
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.
Since G is not connected and H
is connected, G and H cannot be
isomorphic.
Why?
Any graph
isomorphic to H must be connected.
(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.
Define
φ
from C
5
to its complement by
φ
(
a
)=a
,
φ
(
b
)=c
,
φ
(
c
)=e
,
φ
(
d
)=b
,
and
φ
(
e
)=d
.
φ
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 Summer '12
 Rittered
 Graph Theory, Vertex, Planar graph, κ

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