Distances in Sierpi nski Triangle Graphs Sara Sabrina Zemlji c - - PowerPoint PPT Presentation

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Distances in Sierpi nski Triangle Graphs Sara Sabrina Zemlji c - - PowerPoint PPT Presentation

Nov 18, 2022 132 likes •486 views

Distances in Sierpi nski Triangle Graphs Sara Sabrina Zemlji c joint work with Andreas M. Hinz June 18th 2015 Motivation Sierpi nski triangle introduced by Wac law Sierpi nski in 1915. S. S. Zemlji c 1 Motivation S. S.

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Distances in Sierpi´ nski Triangle Graphs

Sara Sabrina Zemljiˇ c

joint work with Andreas M. Hinz

June 18th 2015

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Motivation Sierpi´ nski triangle introduced by Wac law Sierpi´ nski in 1915.

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Motivation

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Motivation

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Motivation

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Motivation

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Motivation Sierpi´ nski triangle introduced by Wac law Sierpi´ nski in 1915. Sierpi´ nski graphs introduced by Klavˇ zar and Milutinovi´ c in 1997, connected to the Tower of Hanoi puzzle – state graphs for the Switching Tower of Hanoi puzzle.

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Motivation

ˆ 02 01 2 00 1 12 11 22 21 ˆ 1 10 20 ˆ 2 002 001 000 012 011 022 021 010 020 102 101 202 201 100 200 112 111 122 121 212 211 222 221 110 120 210 220 000 001 002 010 011 012 020 021 022 100 101 102 110 111 112 120 121 122 200 201 202 210 211 212 220 221 222

Graphs ST 3

3 (left) and S3 3 (right).

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Motivation Sierpi´ nski triangle introduced by Wac law Sierpi´ nski in 1915. Sierpi´ nski graphs introduced by Klavˇ zar and Milutinovi´ c in 1997, connected to the Tower of Hanoi puzzle – state graphs for the Switching Tower of Hanoi puzzle. Applications outside mathemtics

Physics – spectral theory (Laplace operator), spanning trees (Kirchhof’s Theorem), Psychology – ”state graphs” of the Tower of Hanoi puzzle.

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Notations [n] := {1, . . . , n}, [n]0 := {0, . . . , n − 1}, T := [3]0 = {0, 1, 2},

  • T := {ˆ

0, ˆ 1, ˆ 2}, P := [p]0 = {0, . . . , p − 1},

  • P := {ˆ

k | k ∈ P}.

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Definition (Idle peg labeling) Let n ∈ N. Sierpi´ nski triangle graphs ST n

3 ...

... are the graphs defined as follows:

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Definition (Idle peg labeling) Let n ∈ N. Sierpi´ nski triangle graphs ST n

3 ...

... are the graphs defined as follows:

ˆ ˆ 1 ˆ 2 ST 0

3 ∼

= K3 V (ST 0

3 ) =

T

  • vertices ˆ

0, ˆ 1, and ˆ 2 are primitive vertices

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Definition (Idle peg labeling) Let n ∈ N. Sierpi´ nski triangle graphs ST n

3 ...

... are the graphs defined as follows:

V (ST n

3 ) =

T ∪ {s ∈ T ν | ν ∈ [n]} , E(ST n

3 ) =

  • {ˆ

k, kn−1j} | k ∈ T, j ∈ T \ {k}

  • ∪
  • {sk, sj} | s ∈ T n−1, {j, k} ∈ (T

2)

  • ∪
  • {s(3 − i − j)in−1−νk, sj} | s ∈ T ν−1, ν ∈ [n], i ∈ T, j, k ∈ T \ {i}
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Example – Idle peg labeling

ˆ ˆ 1 ˆ 2

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Example – Idle peg labeling

ˆ ˆ 1 ˆ 2 ˆ 2 1 ˆ 1 ˆ 2

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Example – Idle peg labeling

ˆ 02 01 2 00 1 12 11 22 21 ˆ 1 10 20 ˆ 2

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Example – Idle peg labeling

ˆ 02 01 2 00 1 12 11 22 21 ˆ 1 10 20 ˆ 2

002 001 000 012 011 022 021 010 020 102 101 202 201 100 200 112 111 122 121 212 211 222 221 110 120 210 220

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Example – Idle peg labeling

ˆ 02 01 2 00 1 12 11 22 21 ˆ 1 10 20 ˆ 2

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Example – Idle peg labeling

ˆ 02 01 2 00 1 12 11 22 21 ˆ 1 10 20 ˆ 2

000 001 002 010 020 011 012 021 022 100 200 101 102 201 202 110 120 210 220 111 112 121 122 211 212 221 222

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Example – Idle peg labeling

ˆ 02 01 2 00 1 12 11 22 21 ˆ 1 10 20 ˆ 2

002 001 000 012 011 022 021 010 020 102 101 202 201 100 200 112 111 122 121 212 211 222 221 110 120 210 220

000 001 002 010 020 011 012 021 022 100 200 101 102 201 202 110 120 210 220 111 112 121 122 211 212 221 222

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Contraction labeling Let n ∈ N. Contraction labeling of Sierpi´ nski triangle graphs ST n

3

ˆ ˆ 1 ˆ 2 ST 0

3 ∼

= K3 V (ST 0

3 ) =

T

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Contraction labeling Let n ∈ N. Contraction labeling of Sierpi´ nski triangle graphs ST n

3

V (ST n

3 ) =

T ∪

  • s{i, j} | s ∈ T ν−1, ν ∈ [n], {i, j} ∈ (T

2)

  • ,

E(ST n

3 ) =

  • {ˆ

k, kn−1{j, k}} | k ∈ T, j ∈ T \ {k}

  • ∪
  • {s{i, j}, s{i, k}} | s ∈ T n−1, i ∈ T, {j, k} ∈ (T\{i}

2

)

  • ∪
  • {skin−1−ν{i, j}, s{i, k}} | s ∈ T ν−1, ν ∈ [n − 1], i ∈ T, {j, k} ∈ T \ {i}
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Basic properties |ST n

3 | = 3

2(3n + 1) ST n

3 = 3n+1

graphs ST n

3 are connected

ˆ 02 01 2 00 1 12 11 22 21 ˆ 1 10 20 ˆ 2

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Distance to a primitive vertex Lemma. If n ∈ N and ν ∈ [n]0, then for any s, t ∈ V (ST ν

3 )

dn(s, t) = 2n−νdν(s, t) .

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Distance to a primitive vertex

ˆ 02 01 2 00 1 12 11 22 21 ˆ 1 10 20 ˆ 2

002 001 000 012 011 022 021 010 020 102 101 202 201 100 200 112 111 122 121 212 211 222 221 110 120 210 220

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Distance to a primitive vertex Lemma. If n ∈ N and ν ∈ [n]0, then for any s, t ∈ V (ST ν

3 )

dn(s, t) = 2n−νdν(s, t) . Proposition. If ν ∈ N and s ∈ T ν, then d0(ˆ k, ˆ ℓ) = (k = ℓ) , and dν(s, ˆ ℓ) = 1 + (s1 = ℓ) +

ν

∑

d=2

(sd = ℓ) · 2d−1 .a There are 1 + (s1 = ℓ) shortest paths between s and ˆ ℓ.

aHere (X) is Iverson convention, which is 1 if X is true and 0 if X is false.

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Distances – special case Let {i, j, k} = T, n ∈ N and s ∈ T n. dn+1(is, j) = dn(s, ˆ k) dn+1(is, i) = min{dn(s, ˆ k) | k ∈ T \ {i}} + 2n If s = iκsn−κs, κ ∈ [n − 1]0, then dn+1(is, i) = dn(s, sn−κ) + 2n and the shortest path goes through vertex 3 − i − sn−κ. two shortest paths between is and i iff is = iν+1, ν ∈ [n] two shortest paths between is and j iff is = ikν, ν ∈ [n]

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Distances – general formula Theorem. If n ∈ N and ν ∈ [n]0, then for any s ∈ V (ST n

3 ), t ∈ V (ST ν 3 ), and

{i, j, k} = T, dn+1(is, jt) = min{dn(s, ˆ j) + 2n−νdν(t, ˆ i) ; dn(s, ˆ k) + 2n + 2n−νdν(t, ˆ k)} . Problem of two shortest paths: shortest path either goes directly from i-subgraph to j-subgraph, or it goes through k-subgraph. It can also happen that there are two shortest paths.

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Comparison with metric properties of Sierpi´ nski graphs Sn

3

ST n

3

d(ss, st) = d(s, t) dn(s, t) = 2n−νdν(s, t) d(s, jn) =

n

∑

d=1

(sd = j)2d−1 dν(s, ˆ ℓ) = 1 + (s1 = ℓ) +

ν

∑

d=2

(sd = ℓ)2d−1 d(is, jt) = min{ddir(is, jt), dindir(is, jt)} diam(Sn

3 ) = 2n − 1

diam(ST n

3 ) = 2n

∑

i∈T

d(s, in) = 2n+1 − 2

∑

i∈T

d(s, in) = 2n+1

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Automaton

(1, ·), (·, 0), (0, 1) ({0, 1}, ·), (·, {0, 1}), (0, {1, 2}) (·, {0, 2}), ({1, 2}, ·) (2, 2) (2, {1, 2}) ({0, 2}, {1, 2}) (0, 0) (1, 1) (1, 0), (1, {0, 1}), ({0, 1}, {0, 1}) (1, {0, 2}), ({0, 1}, {0, 2}) (0, {0, 1}), ({1, 2}, {0, 1}) (2, ∅), ({0, 2}, ∅) (2, 1) (0, 2) (2, 2), (2, {1, 2}), ({0, 2}, {1, 2}) (2, {0, 2}), ({0, 2}, {0, 2}) (0, {1, 2}), ({1, 2}, {1, 2}) (1, ∅), ({0, 1}, ∅) (1, 0) (1, {0, 1}) ({0, 1}, {0, 1}) (2, ·), (·, 2), (0, 1) ({0, 2}, ·), (·, {1, 2}), (0, {0, 1}) (·, {0, 2}), ({1, 2}, ·) (0, 2) (2, 1) (0, 1), (0, {0, 2}) ({1, 2}, {0, 2}) (2, 0), (2, {0, 1}) ({0, 2}, {0, 1}) (1, 2), (1, {1, 2}) ({0, 1}, {1, 2}) (0, ∅), ({1, 2}, ∅) (1, 1) (0, 0)

A B C D E Example

d4(002{0, 2}, 112{1, 2}) = 16 (direct) d4(020{1, 2}, 12{0, 2}) = 13 (two shortest paths) d4(022{0, 1}, 12{0, 2}) = 12 (indirect)

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Sierpi´ nski triangle graphs ST n

p

Jakovac, A 2-parametric generalization of Sierpi´ nski gasket graphs,

  • Ars. Combin. 116 (2014) 395–405.

Sierpi´ nski triangle graphs ST n

p (n ∈ N) ...

... are the graphs defined by:

V (ST n

p ) =

P ∪

  • s{i, j} | s ∈ Pν−1, ν ∈ [n], {i, j} ∈ (P

2)

  • ,

E(ST n

p ) =

  • {ˆ

k, kn−1{j, k}} | k ∈ P, j ∈ P \ {k}

  • ∪
  • {s{i, j}, s{i, k}} | s ∈ Pn−1, i ∈ P, {j, k} ∈ (P\{i}

2

)

  • ∪
  • {skin−1−ν{i, j}, s{i, k}} | s ∈ Pν−1, ν ∈ [n − 1], i ∈ P, {j, k} ∈ P \ {i}
  • .

As before, ST 0

p ∼

= Kp and V (ST 0

p ) =

P.

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Example ST 1

4 {0, 1} {0, 2} ˆ {0, 3} ˆ 3 {2, 3} {1, 3} ˆ 2 ˆ 1 {1, 2}

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Distances in ST n

p

dν(s{i, j}, ˆ ℓ) = 1 + (i = ℓ)(j = ℓ) +

ν−1

∑

d=1

(sd = ℓ) · 2d ∈ [2ν] there are 1 + (p − 2)(i = ℓ)(j = ℓ) s{i, j}, ˆ ℓ-shortest paths diam(ST n

p ) = 2n

∀ s ∈ V (ST n

p ) : p−1

∑

ℓ=0

dn(s, ˆ ℓ) = (p − 1) · 2n dn+1(is, jt) = min{dn(s, ˆ j) + 2n−νdν(t, ˆ i) ; dn(s, ˆ k) + 2n−νdν(t, ˆ k) + 2n | k ∈ P \ {i, j}} .

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Open Problems explicit formula for average distance

  • ther metric properties which are known for Sierpi´

nski graphs Sn

p

we are currently working on the decision automaton for p > 3

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THANK YOU!