Near Automorphisms of Graphs

陳伯亮 (Bor-Liang Chen)

台中技術學院2009 年 7 月 29 日

• Let f be a permutation of V (G).

• Let f(x,y) = |dG(x,y)-dG(f(x),f(y))| for all the unorder

ed pairs {x,y} of distinct vertices of G.

• The total relative displacement of permutation f in

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 t

he total relative displacement of G.

• The permutation f with f(G) = (G) is called a near

automorphism of G

(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)

Known results

Theorem. (Reid, 2002)

• Lemma.

f(G) and (G) are even.

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.

Some Results

• Lemma.

If G is not a complete graph, then

.

• Lemma.

If G is not a complete graph, then

.

G2

4)(2 GVG

• Theorem.

If G is not a complete graph, then

.

• Lemma.

If , then G is a bipartite graph.

4)(22 GVG

4)(2 GVG

Graphs with (G) = 2|V(G)|4

• Paths

• Even cycles

• Some Trees

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}.

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, t

hen (G) = 2.

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, t

hen (G) = 2.

• The near automorphism may be chosen as f = (uv).

i

k

j

x y

w f(k)

f(i)

f(j)

f(z) f(w)

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)

z

f(y)

f(x)

i

k

j

x y

zf(k)

f(i)

f(j)

F(z) F(y)

f(x)

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)

• Property.

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

},{)}(),({,2

},{)}(),({,0

vuvfuf

vuvfufuvGf

v

u

u

v