Near automorphisms of graphs
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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 .

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Near Automorphisms of Graphs

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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


Near automorphisms of graphs

  • 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


Near automorphisms of graphs

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)


Theorem reid 2002

Theorem. (Reid, 2002)


Near automorphisms of graphs

Some Results

  • Lemma.

    f(G) and (G) are even.


Near automorphisms of graphs

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.


Near automorphisms of graphs

  • Lemma.

    If G is not a complete graph, then

    .

  • Lemma.

    If G is not a complete graph, then

    .


Near automorphisms of graphs

  • 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}.


Near automorphisms of graphs

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.


Near automorphisms of graphs

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)

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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)

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Near automorphisms of graphs

  • Property.

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


Near automorphisms of graphs

u

v


Near automorphisms of graphs

u

v


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