This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1. An automorphism of a graph G is an isomorphism of G with itself, i.e., a 1to1 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 . 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 4cycle, 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 nonisomorphic graphs with n vertices. Prove that summationdisplay G ∈ Graphs n n !...
View
Full
Document
This note was uploaded on 07/15/2010 for the course MACM 201 taught by Professor Marnimishna during the Spring '09 term at Simon Fraser.
 Spring '09
 MarniMishna

Click to edit the document details