Near Automorphisms of Graphs
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Transcript of Near Automorphisms of Graphs
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