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

Distances in Sierpi´ nski Triangle Graphs Sara Sabrina Zemljiˇ c joint work with Andreas M. Hinz June 18th 2015

Motivation Sierpi´ nski triangle introduced by Wac� law Sierpi´ nski in 1915. S. S. Zemljiˇ c 1

Motivation S. S. Zemljiˇ c 1

Motivation S. S. Zemljiˇ c 1

Motivation S. S. Zemljiˇ c 1

Motivation S. S. Zemljiˇ c 1

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. S. S. Zemljiˇ c 1

Motivation ˆ 0 000 002 001 02 01 001 002 000 011 012 021 010 020 022 2 1 011 022 010 00 020 012 021 102 101 202 201 100 200 12 11 22 21 101 102 201 202 100 200 111 121 211 112 221 110 120 210 220 122 212 222 ˆ 110 10 120 0 210 20 220 ˆ 1 2 111 112 121 122 211 212 221 222 Graphs ST 3 3 (left) and S 3 3 (right). S. S. Zemljiˇ c 1

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. S. S. Zemljiˇ c 1

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 } . S. S. Zemljiˇ c 2

Definition (Idle peg labeling) Let n ∈ N . nski triangle graphs ST n Sierpi´ 3 ... ... are the graphs defined as follows: S. S. Zemljiˇ c 3

Definition (Idle peg labeling) Let n ∈ N . nski triangle graphs ST n Sierpi´ 3 ... ... are the graphs defined as follows: ˆ 0 3 ∼ ST 0 = K 3 3 ) = � V ( ST 0 T ˆ ˆ 1 2 • vertices ˆ 0, ˆ 1, and ˆ 2 are primitive vertices S. S. Zemljiˇ c 3

Definition (Idle peg labeling) Let n ∈ N . nski triangle graphs ST n Sierpi´ 3 ... ... are the graphs defined as follows: T ∪ { s ∈ T ν | ν ∈ [ n ] } , V ( ST n 3 ) = � � � � � k , k n − 1 j } | k ∈ T , j ∈ T \ { k } { sk , sj } | s ∈ T n − 1 , { j , k } ∈ ( T E ( ST n { ˆ 3 ) = ∪ 2 ) � � { s ( 3 − i − j ) i n − 1 − ν k , sj } | s ∈ T ν − 1 , ν ∈ [ n ] , i ∈ T , j , k ∈ T \ { i } ∪ S. S. Zemljiˇ c 3

Example – Idle peg labeling ˆ 0 ˆ ˆ 1 2 S. S. Zemljiˇ c 4

Example – Idle peg labeling ˆ ˆ 0 0 2 1 ˆ ˆ 0 ˆ ˆ 1 1 2 2 S. S. Zemljiˇ c 4

Example – Idle peg labeling ˆ 0 02 01 2 1 00 12 11 22 21 ˆ 10 0 20 ˆ 1 2 S. S. Zemljiˇ c 4

Example – Idle peg labeling ˆ 0 002 001 02 01 000 011 012 021 022 2 1 00 010 020 102 101 202 201 12 11 22 21 100 200 111 121 211 112 221 122 212 222 ˆ 10 0 20 ˆ 1 110 120 210 220 2 S. S. Zemljiˇ c 4

Example – Idle peg labeling ˆ 0 02 01 2 1 00 12 11 22 21 ˆ 10 0 20 ˆ 1 2 S. S. Zemljiˇ c 4

Example – Idle peg labeling ˆ 0 000 001 002 02 01 010 020 011 022 012 021 2 1 00 100 200 101 102 201 202 12 11 22 21 110 120 210 220 111 222 112 121 122 211 212 221 ˆ 10 0 20 ˆ 1 2 S. S. Zemljiˇ c 4

Example – Idle peg labeling ˆ 0 000 002 001 001 002 02 01 000 010 020 011 012 021 022 011 022 012 021 2 1 00 010 020 100 200 102 101 202 201 101 102 201 202 12 11 22 21 100 200 110 120 210 220 111 121 211 112 221 122 212 222 111 222 112 121 122 211 212 221 ˆ 10 0 20 ˆ 1 110 120 210 220 2 S. S. Zemljiˇ c 4

Contraction labeling Let n ∈ N . nski triangle graphs ST n Contraction labeling of Sierpi´ 3 ˆ 0 3 ∼ ST 0 = K 3 V ( ST 0 3 ) = � T ˆ ˆ 1 2 S. S. Zemljiˇ c 5

Contraction labeling Let n ∈ N . nski triangle graphs ST n Contraction labeling of Sierpi´ 3 � � V ( ST n 3 ) = � s { i , j } | s ∈ T ν − 1 , ν ∈ [ n ] , { i , j } ∈ ( T T ∪ 2 ) , � � E ( ST n k , k n − 1 { j , k }} | k ∈ T , j ∈ T \ { k } { ˆ 3 ) = ∪ � � { s { i , j } , s { i , k }} | s ∈ T n − 1 , i ∈ T , { j , k } ∈ ( T \{ i } ) ∪ 2 � � { ski n − 1 − ν { i , j } , s { i , k }} | s ∈ T ν − 1 , ν ∈ [ n − 1 ] , i ∈ T , { j , k } ∈ T \ { i } S. S. Zemljiˇ c 5

Basic properties 3 | = 3 2 ( 3 n + 1 ) | ST n ˆ 0 3 � = 3 n + 1 � ST n 02 01 graphs ST n 3 are connected 2 1 00 12 11 22 21 ˆ 1 10 0 20 2 ˆ S. S. Zemljiˇ c 6

Distance to a primitive vertex Lemma. If n ∈ N and ν ∈ [ n ] 0 , then for any s , t ∈ V ( ST ν 3 ) d n ( s , t ) = 2 n − ν d ν ( s , t ) . S. S. Zemljiˇ c 7

Distance to a primitive vertex ˆ 0 002 001 02 01 000 011 012 021 022 2 1 00 010 020 102 101 202 201 12 11 22 21 100 200 111 121 211 112 221 122 212 222 ˆ 10 0 20 ˆ 1 110 120 210 220 2 S. S. Zemljiˇ c 7

Distance to a primitive vertex Lemma. If n ∈ N and ν ∈ [ n ] 0 , then for any s , t ∈ V ( ST ν 3 ) d n ( s , t ) = 2 n − ν d ν ( s , t ) . Proposition. If ν ∈ N and s ∈ T ν , then d 0 ( ˆ k , ˆ ℓ ) = ( k � = ℓ ) , and ν ( s d � = ℓ ) · 2 d − 1 . a d ν ( s , ˆ ∑ ℓ ) = 1 + ( s 1 = ℓ ) + d = 2 There are 1 + ( s 1 = ℓ ) shortest paths between s and ˆ ℓ . a Here (X) is Iverson convention, which is 1 if X is true and 0 if X is false. S. S. Zemljiˇ c 7

Distances – special case Let { i , j , k } = T , n ∈ N and s ∈ T n . d n + 1 ( is , j ) = d n ( s , ˆ k ) d n + 1 ( is , i ) = min { d n ( s , ˆ k ) | k ∈ T \ { i }} + 2 n s n − κ ) + 2 n and the If s = i κ s n − κ s , κ ∈ [ n − 1 ] 0 , then d n + 1 ( is , i ) = d n ( s , � shortest path goes through vertex 3 − i − s n − κ . two shortest paths between is and i iff is = i ν + 1 , ν ∈ [ n ] two shortest paths between is and j iff is = ik ν , ν ∈ [ n ] S. S. Zemljiˇ c 8

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 , k ) + 2 n + 2 n − ν d ν ( t , ˆ j ) + 2 n − ν d ν ( t , ˆ d n + 1 ( is , jt ) = min { d n ( s , ˆ i ) ; d n ( s , 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. S. S. Zemljiˇ c 9

Comparison with metric properties of Sierpi´ nski graphs S n ST n 3 3 d n ( s , t ) = 2 n − ν d ν ( s , t ) d ( ss , st ) = d ( s , t ) n ν ( s d � = j ) 2 d − 1 ( s d � = ℓ ) 2 d − 1 d ( s , j n ) = d ν ( s , ˆ ∑ ∑ ℓ ) = 1 + ( s 1 = ℓ ) + d = 1 d = 2 d ( is , jt ) = min { d dir ( is , jt ) , d indir ( is , jt ) } 3 ) = 2 n − 1 diam ( S n diam ( ST n 3 ) = 2 n d ( s , i n ) = 2 n + 1 − 2 d ( s , i n ) = 2 n + 1 ∑ ∑ i ∈ T i ∈ T S. S. Zemljiˇ c 10

Automaton ( 0, 2 ) ( 2, 1 ) A ( { 0, 1 } , · ) , ( · , { 0, 1 } ) , ( 0, { 1, 2 } ) ( 1, · ) , ( · , 0 ) , ( 0, 1 ) ( · , { 0, 2 } ) , ( { 1, 2 } , · ) Example ( 2, 2 ) ( 0, 0 ) D d 4 ( 002 { 0, 2 } , 112 { 1, 2 } ) = 16 ( 2, { 1, 2 } ) ( 1, 1 ) ( 1, 0 ) , ( 1, { 0, 1 } ) , ( { 0, 1 } , { 0, 1 } ) ( { 0, 2 } , { 1, 2 } ) (direct) ( 1, { 0, 2 } ) , ( { 0, 1 } , { 0, 2 } ) ( 0, { 0, 1 } ) , ( { 1, 2 } , { 0, 1 } ) ( 0, 1 ) , ( 0, { 0, 2 } ) ( 2, ∅ ) , ( { 0, 2 } , ∅ ) d 4 ( 020 { 1, 2 } , 12 { 0, 2 } ) = 13 ( { 1, 2 } , { 0, 2 } ) ( 2, 0 ) , ( 2, { 0, 1 } ) (two shortest paths) ( { 0, 2 } , { 0, 1 } ) B ( 2, 2 ) , ( 2, { 1, 2 } ) , ( { 0, 2 } , { 1, 2 } ) ( 1, 2 ) , ( 1, { 1, 2 } ) ( { 0, 1 } , { 1, 2 } ) ( 2, { 0, 2 } ) , ( { 0, 2 } , { 0, 2 } ) ( 0, ∅ ) , ( { 1, 2 } , ∅ ) d 4 ( 022 { 0, 1 } , 12 { 0, 2 } ) = 12 ( 0, { 1, 2 } ) , ( { 1, 2 } , { 1, 2 } ) ( 1, ∅ ) , ( { 0, 1 } , ∅ ) (indirect) ( 1, 0 ) ( 2, 1 ) E ( 1, { 0, 1 } ) ( 0, 2 ) ( { 0, 1 } , { 0, 1 } ) ( 2, · ) , ( · , 2 ) , ( 0, 1 ) ( { 0, 2 } , · ) , ( · , { 1, 2 } ) , ( 0, { 0, 1 } ) ( · , { 0, 2 } ) , ( { 1, 2 } , · ) C ( 1, 1 ) ( 0, 0 ) S. S. Zemljiˇ c 11

nski triangle graphs ST n Sierpi´ p Jakovac, A 2-parametric generalization of Sierpi´ nski gasket graphs , Ars. Combin. 116 (2014) 395–405. nski triangle graphs ST n p ( n ∈ N ) ... Sierpi´ ... are the graphs defined by: � � V ( ST n p ) = � s { i , j } | s ∈ P ν − 1 , ν ∈ [ n ] , { i , j } ∈ ( P P ∪ 2 ) , � � E ( ST n k , k n − 1 { j , k }} | k ∈ P , j ∈ P \ { k } { ˆ p ) = ∪ � � { s { i , j } , s { i , k }} | s ∈ P n − 1 , i ∈ P , { j , k } ∈ ( P \{ i } ) ∪ 2 � � { ski n − 1 − ν { i , j } , s { i , k }} | s ∈ P ν − 1 , ν ∈ [ n − 1 ] , i ∈ P , { j , k } ∈ P \ { i } . p ∼ p ) = � As before, ST 0 = K p and V ( ST 0 P . S. S. Zemljiˇ c 12

Example ST 1 4 { 0, 1 } { 0, 2 } ˆ 0 { 0, 3 } ˆ 3 { 1, 3 } { 2, 3 } ˆ ˆ 1 2 { 1, 2 } S. S. Zemljiˇ c 13