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Exercise Set 1
.
1 b) The shortest
a

g
path has length 2; the longest has length 8. Of the 14
a

g
paths:
1 has length 2, 1 has length 3, 1 has length 4, 2 have length 5, 4 have length 6, 3 have
length 7, and 2 have length 8.
(c) There are 7
a

g
walks of length 4. Only 1 of these is a path.
(d) There are 22 paths of length 2 in
G
, if the reverse of a path is regarded as the same
path (i.e.
a, b, c
is the same as
c, b, a
). Otherwise there are 44.
(e) 3 :
a, b, c, a
; 4 :
a, c, d, b, a
; 5 :
b, d, e, f, g, b
; 6 :
b, d, e, f, h, g, b
; 7 :
a, c, d, e, f, g, b, a
;
8 :
a, c, d, e, f, h, g, b, a.
2 (a)
(
n
2
)
(b)
n
(c)
n

1
(d)
mn
(e) 12
3. For any
n
≥
2 the graph
P
n
has no cycles, but has the closed walk
v
0
, v
1
, . . . , v
n
, v
n

1
, v
n

2
. . . , v
0
containing every vertex.
4. Consider a shortest closed trail of positive length containing
x
, say
x
0
, x
1
, x
2
, . . . , x
n
, x
0
where
x
0
=
x
. If this is a cycle, then there is a cycle containing
x
. Otherwise it is not
a cycle, so it contains a repeated vertex. Thus there exists integers
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This note was uploaded on 02/05/2011 for the course MATH 377 taught by Professor Stephenlang during the Spring '11 term at University of Victoria.
 Spring '11
 stephenlang
 Calculus

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