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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1. An automorphism of a graph G is an isomorphism of G with itself, i.e., a 1-to-1 function : V ( G ) V ( G ) such that { u, v } is an edge of G iff { ( 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 such that ( 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 !...
View Full Document

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online