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# Assn8 - n vertices Prove that X G ∈ Graphs n n aut G = 2...

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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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