Section 48
1.
Let
G
be the graph in the °gure.
(a)
How many di/erent paths are there from
a
to
b
?
Each path has to go through that point
w
in the middle (the waist
of the graph). There are
5
paths from
a
to
w
and
3
paths from
w
to
b
, so there are
15 = 3
°
5
paths from
a
to
b
.
(b)
How many di/erent walks are there from
a
to
b
.
In general, if there is any walk between two vertices, there are
in°nitely many, because you can just walk back and forth. So we
could go from
a
to
w
to
b
, back to
w
, back to
b
back to
w
, back
to
b
, and so on.
4.
Let
n
±
2
be an integer. Form a graph
G
n
whose vertices are all the two
element subsets of
f
1
;
2
; : : : ; n
g
.
In this graph we have an edge between
distinct vertices
f
a; b
g
and
f
c; d
g
exactly when
f
a; b
g \ f
c; d
g
=
;
.
(a)
How many vertices does
G
n
have
?
A vertex is a two element subset of
f
1
;
2
; : : : ; n
g
so there are
°
n
2
±
=
n
(
n
²
1)
=
2
of them.
(b)
How many edges does
G
n
have
?
We have to count the number of ways we can choose two disjoint
vertices. We can pick the °rst vertex in any of
°
n
2
±
ways. Hav
ing picked the °rst vertex, we can pick the second one in any of
°
n
²
2
2
±
ways (we can±t reuse any of the number we used in the
°rst vertex because the two vertices must be disjoint).
So the
number
°
n
2
±°
n
²
2
2
±
counts how many ordered pairs of disjoint vertices we can form.
We have to divide this by
2
because the edges correspond to an
unordered
pairs of disjoint vertices. So the answer is
°
n
2
±°
n
²
2
2
±
2
1
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Notice that this number is zero for
n
= 2
and
n
= 3
. For
n
= 4
it
is
6
°
1
=
2 = 3
and for
n
= 5
it is
10
°
3
=
2 = 15
.
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 Spring '08
 STAFF
 Graph Theory, Ing, JV, 10 K

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