Assn8Soln

# Assn8Soln - 1 An automorphism of a graph G is an...

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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 ) ( 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 . The identity (a function ϕ such that ϕ ( v ) = v for each v V ( G )) is always an automorphism, thus at least one automorphism al- ways exists. The smallest graph with exactly one automorphism is the following: 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 . Each row in the following table defines an automorphism (the value in the first column is the image of a , the value in the second one is the image of b , etc.); aut( G ) = 8. a b c d b c d a c d a b d a b c a d c b d c b a c b a d b a d c Let Graphs n be the set of all mutually non-isomorphic graphs with n vertices. Prove that summationdisplay G Graphs n n !

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