near automorphisms of graphs
Download
Skip this Video
Download Presentation
Near Automorphisms of Graphs

Loading in 2 Seconds...

play fullscreen
1 / 17

Near Automorphisms of Graphs - PowerPoint PPT Presentation


  • 101 Views
  • Uploaded on

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 ) = |d G ( x,y ) -d G ( f ( x ) ,f ( y )) | for all the unordered pairs { x,y } of distinct vertices of G .

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Near Automorphisms of Graphs' - tyne


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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

slide2
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
slide3
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)

slide5
Some Results
  • Lemma.

f(G) and (G) are even.

slide6
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.

slide7
Lemma.

If G is not a complete graph, then

.

  • Lemma.

If G is not a complete graph, then

.

slide8
Theorem.

If G is not a complete graph, then

.

  • Lemma.

If , then G is a bipartite graph.

graphs with g 2 v g 4
Graphs with (G) = 2|V(G)|4
  • Paths
  • Even cycles
  • Some Trees
graphs with g 2
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}.

slide11
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.
slide12
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).
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
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)

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 l1
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)

slide15
Property.

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

slide16
u

v

slide17
u

v

ad