Bivariegated Graphs And Their Isomorphisms
Bivariegated Graphs And Their Isomorphisms
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The set of all automorphisms of G forms a group, which will be denoted Aut (G) .
It is an interesting problem to look at the relationship between the automorphism groups of the bivariegated graph G = G, - f - G 2 and its factors, G = (V ,E ) and G 2 = (V 2 ,E 2 ). For, suppose h is an automorphism of one of the factors of G, say G^. Then, unless h is the identity map, h maps at least one vertex, say u, to another. Because G is bivariegated, u is edged with some vertex v in G 2 . Now, in order 4...1 42 for h to correspond to an automorphism on the entire araph G, h must similarly move v. This motivates the followinq theorem: Theorem 4.2 : Let g be any automorphism of one of the factors of G = G, - f - G 2 (without loss of generality, the factor G ) . If g is an automorphism on G that agrees with g on G (that is, g restricted to G, is identically g) , then it must be true that for any vertex v e V- , g(v) = f (g(f _1 (v))) .
Proof: If g is an automorphism of G, then it preserves edges in G.
What to read after Bivariegated Graphs And Their Isomorphisms?
You can find similar books in the "Read Also" column, or choose other free books by Riddle, Fay Aycock to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
The set of all automorphisms of G forms a group, which will be denoted Aut (G) .
It is an interesting problem to look at the relationship between the automorphism groups of the bivariegated graph G = G, - f - G 2 and its factors, G = (V ,E ) and G 2 = (V 2 ,E 2 ). For, suppose h is an automorphism of one of the factors of G, say G^. Then, unless h is the identity map, h maps at least one vertex, say u, to another. Because G is bivariegated, u is edged with some vertex v in G 2 . Now, in order 4...1 42 for h to correspond to an automorphism on the entire araph G, h must similarly move v. This motivates the followinq theorem: Theorem 4.2 : Let g be any automorphism of one of the factors of G = G, - f - G 2 (without loss of generality, the factor G ) . If g is an automorphism on G that agrees with g on G (that is, g restricted to G, is identically g) , then it must be true that for any vertex v e V- , g(v) = f (g(f _1 (v))) .
Proof: If g is an automorphism of G, then it preserves edges in G.
What to read after Bivariegated Graphs And Their Isomorphisms?
You can find similar books in the "Read Also" column, or choose other free books by Riddle, Fay Aycock to read onlineMoreLess
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