Near Automorphisms of Graphs

1. Near Automorphisms of Graphs 陳伯亮 (Bor-Liang Chen) 台中技術學院 2009年7月29日

2. Let f be a permutation of V (G). • Let f(x,y) = |dG(x,y)-dG(f(x),f(y))| for all the unordered pairs {x,y}of distinct vertices of G. • The total relative displacement of permutation fin G is defined to be the value f(G) = f(x,y). • The smallest positive value of f(G) among all the permutations f of V(G) is denoted by (G), called the total relative displacement of G. • The permutation f with f(G) = (G) is called a near automorphism of G

3. Known results • (G) is determined. Paths (Aitken, 1999) Complete partite Graphs (Reid, 2002) Cycles (Chang, Chen and Fu, 2008) • Characterization of trees T (T) = 2 (Chang and Fu, 2007) (T) = 4 (Chang and Fu, 2007)

4. Theorem. (Reid, 2002)

5. Some Results • Lemma. f(G) and (G) are even.

6. Some Results • Lemma. f(G) and (G) are even. {dG(x,y)-dG(f(x),f(y))} = 0   f(G) = f(x,y) = |dG(x,y)-dG(f(x),f(y))| is even.

7. Lemma. If G is not a complete graph, then . • Lemma. If G is not a complete graph, then .

8. Theorem. If G is not a complete graph, then . • Lemma. If , then G is a bipartite graph.

9. Graphs with (G) = 2|V(G)|4 • Paths • Even cycles • Some Trees

10. Graphs with (G) = 2 Theorem. A graph G is of (G) = 2 if and only if there is a near automophism f such that there are two pairs {i,j}, {l,k} such that d(i,j) = 1 = d(f(l),f(k)) and d(l,k) = 2 = d(f(i),f(j)) and d(x,y) = d(f(x),f(y)) for the other unordered paired {x,y}.

11. Property. If there are two vertices u and v of graph G such that deg(u) = deg(v)+1, N(u)-N[v] = {w}, d(v,w) = 2 and dG(x,w)  dG(x,v)-1 for all x  w, then (G) = 2.

12. Property. If there are two vertices u and v of graph G such that deg(u) = deg(v)+1, N(u)-N[v] = {w}, d(v,w) = 2 and dG(x,w)  dG(x,v)-1 for all x  w, then (G) = 2. • The near automorphism may be chosen as f = (uv).

13. d(i,j) = 2 = d(f(l),f(k)) and d(l,k) = 2 = d(f(i),f(j))|{i,j,k,l}| = 3 (Assume j = l) k z f(i) f(x) i j w f(k) f(j) f(y) x y f(z) f(w)

14. d(i,j) = 2 = d(f(l),f(k)) and d(l,k) = 2 = d(f(i),f(j))|{i,j,k,l}| = 3 (Assume j = l) k f(i) i f(k) j z f(x) f(j) x y F(z) F(y)

15. Property. Let the graph G be of diameter 2 and f an automorphism of G. If uv is not edge of G, then

16. u v

17. u v