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1. An
automorphism
of a graph
G
is an isomorphism of
G
with itself,
i.e., a 1-to-1 function
ϕ
:
V
(
G
)
→
V
(
G
) such that
{
u, v
}
is an edge
of
G
iff
{
ϕ
(
u
)
, ϕ
(
v
)
}
is an edge of
G
.
Let aut(
G
) be the number of
automorphisms of
G
.
•
Show that aut(
G
)
≥
1 for every graph
G
. Find a graph with at
least two vertices for that aut(
G
) = 1.
•
Let
G
be a 4-cycle, i.e., a graph with 4 vertices
a
,
b
,
c
and
d
and 4
edges
{
a, b
}
,
{
b, c
}
,
{
c, d
}
and
{
a, d
}
. Determine aut(
G
) and list
all automorphisms of
G
.
•
Let Graphs
n
be the set of all mutually non-isomorphic graphs with
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Unformatted text preview: n vertices. Prove that X G ∈ Graphs n n ! aut( G ) = 2 ( n 2 ) . Hint: modify the proof that there are at most 2 ( n 2 ) n ! mutually non-isomorphic graphs on n vertices. 2. Chapter 9.2, exercises 9 and 11. 3. Chapter 11.3, exercises 20, 21, 22 and 29. 4. Chapter 11.4, exercises 2, 14, 17 and 19. 5. Chapter 11.5, exercises 1 and 3....
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- Spring '09
- MarniMishna
- Morphism, Isomorphism, G. Let Aut, Graphsn, mutually non-isomorphic graphs
-
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