Distances in Sierpiński Triangle Graphs

Distances in Sierpiński Triangle Graphs Sara Sabrina Zemljič joint work with Andreas M. Hinz June 18th 2015

Motivation Sierpiński triangle introduced by Wac law Sierpiński in 1915. S. S. Zemljič 1

Motivation S. S. Zemljič 1

Motivation S. S. Zemljič 1

Motivation S. S. Zemljič 1

Motivation S. S. Zemljič 1

Motivation Sierpiński triangle introduced by Wac law Sierpiński in 1915. Sierpiński graphs introduced by Klavžar and Milutinović in 1997, connected to the Tower of Hanoi puzzle state graphs for the Switching Tower of Hanoi puzzle. S. S. Zemljič 1

Motivation ˆ0 002 001 02 01 000 012 011 022 021 000 001 002 010 020 2 010 00 020 1 011 012 021 022 102 101 202 201 100 200 12 11 22 21 100 200 112 111 122 121 212 211 222 221 110 101 102 120 210 201 202 220 ˆ1 110 10 120 0 210 20 220 ˆ2 111 112 121 122 Graphs ST 3 3 (left) and S 3 3 (right). 211 212 221 222 S. S. Zemljič 1

Motivation Sierpiński triangle introduced by Wac law Sierpiński in 1915. Sierpiński graphs introduced by Klavžar and Milutinović 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. Zemljič 1

Notations [n] := {1,..., n}, [n] 0 := {0,..., n 1}, T := [3] 0 = {0, 1, 2}, T := {ˆ0, ˆ1, ˆ2}, P := [p] 0 = {0,..., p 1}, P := { ˆk k P}. S. S. Zemljič 2

Definition (Idle peg labeling) Let n N. Sierpiński triangle graphs ST n 3...... are the graphs defined as follows: S. S. Zemljič 3

Definition (Idle peg labeling) Let n N. Sierpiński triangle graphs ST n 3...... are the graphs defined as follows: ˆ0 ST 0 3 = K 3 V (ST 0 3 ) = T ˆ1 ˆ2 vertices ˆ0, ˆ1, and ˆ2 are primitive vertices S. S. Zemljič 3

Definition (Idle peg labeling) Let n N. Sierpiński triangle graphs ST n 3...... are the graphs defined as follows: V (ST3 n ) = T {s T ν ν [n]}, { } E (ST3 n ) = { ˆk, k n 1 j} k T, j T \ {k} } {{sk, sj} s T n 1, {j, k} ( T 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. Zemljič 3

Example Idle peg labeling ˆ0 ˆ1 ˆ2 S. S. Zemljič 4

Example Idle peg labeling ˆ0 2 1 ˆ1 0 ˆ2 S. S. Zemljič 4

Example Idle peg labeling ˆ0 02 01 2 00 1 12 11 22 21 ˆ1 10 0 20 ˆ2 S. S. Zemljič 4

Example Idle peg labeling ˆ0 002 001 02 000 01 012 011 022 021 2 010 00 020 1 102 101 202 201 12 11 22 21 100 200 112 111 122 121 212 211 222 221 ˆ1 110 10 120 0 210 20 220 ˆ2 S. S. Zemljič 4

Example Idle peg labeling ˆ0 02 01 2 00 1 12 11 22 21 ˆ1 10 0 20 ˆ2 S. S. Zemljič 4

Example Idle peg labeling ˆ0 000 001 002 02 01 010 020 2 011 022 012 021 00 100 200 1 101 102 201 202 12 11 22 21 110 120 210 220 111 112 121 122 211 212 221 222 ˆ1 10 0 20 ˆ2 S. S. Zemljič 4

Example Idle peg labeling ˆ0 000 002 001 02 001 002 000 01 012 010 020 011 022 021 2 011 022 012 021 010 00 020 1 100 200 102 101 202 201 12 101 102 201 202 11 22 21 100 200 112 110 120 210 220 111 121 211 122 212 222 221 111 112 121 122 211 212 221 222 ˆ1 110 10 120 0 210 20 220 ˆ2 S. S. Zemljič 4

Contraction labeling Let n N. Contraction labeling of Sierpiński triangle graphs ST n 3 ˆ0 ST 0 3 = K 3 V (ST 0 3 ) = T ˆ1 ˆ2 S. S. Zemljič 5

Contraction labeling Let n N. Contraction labeling of Sierpiński triangle graphs ST n 3 } V (ST3 n ) = T {s{i, j} s T ν 1, ν [n], {i, j} ( T 2 ), { } E (ST3 n ) = { ˆk, k n 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 ) { } {ski n 1 ν {i, j}, s{i, k}} s T ν 1, ν [n 1], i T, {j, k} T \ {i} S. S. Zemljič 5

Basic properties ST n 3 = 3 2 (3n + 1) ST n 3 = 3 n+1 ˆ0 02 01 graphs ST n 3 are connected 2 00 1 12 11 22 21 ˆ1 10 0 20 ˆ2 S. S. Zemljič 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. Zemljič 7

Distance to a primitive vertex ˆ0 002 001 02 01 000 012 011 022 021 2 010 00 020 1 102 101 202 201 12 11 22 21 100 200 112 111 122 121 212 211 222 221 ˆ1 110 10 120 0 210 20 220 ˆ2 S. S. Zemljič 7

Distance to a primitive vertex Lemma. If n N and ν [n] 0, then for any s, t V (ST3 ν) d n (s, t) = 2 n ν d ν (s, t). Proposition. If ν N and s T ν, then d 0 ( ˆk, ˆl) = (k = l), and d ν (s, ˆl) = 1 + (s 1 = l) + ν d=2 (s d = l) 2 d 1. a There are 1 + (s 1 = l) shortest paths between s and ˆl. a Here (X) is Iverson convention, which is 1 if X is true and 0 if X is false. S. S. Zemljič 7

Distances special case Let {i, j, k} = T, n N and s T n. d n+1 (is, j) = d n (s, ˆk) d n+1 (is, i) = min{d n (s, ˆk) k T \ {i}} + 2 n If s = i κ s n κ s, κ [n 1] 0, then d n+1 (is, i) = d n (s, s n κ ) + 2 n and the 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. Zemljič 8

Distances general formula Theorem. If n N and ν [n] 0, then for any s V (ST3 n), t V (ST 3 ν ), and {i, j, k} = T, d n+1 (is, jt) = min{d n (s, ĵ) + 2 n ν d ν (t, î) ; d n (s, ˆk) + 2 n + 2 n ν d ν (t, ˆk)}. 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. Zemljič 9

Comparison with metric properties of Sierpiński graphs S n 3 ST n 3 d(ss, st) = d(s, t) d n (s, t) = 2 n ν d ν (s, t) d(s, j n ) = n d=1 diam(s n 3 ) = 2n 1 d(s, i n ) = 2 n+1 2 i T (s d = j)2 d 1 d ν (s, ˆl) = 1 + (s 1 = l) + d(is, jt) = min{d dir (is, jt), d indir (is, jt)} ν d=2 diam(st n 3 ) = 2n d(s, i n ) = 2 n+1 i T (s d = l)2 d 1 S. S. Zemljič 10

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

Sierpiński triangle graphs ST n p Jakovac, A 2-parametric generalization of Sierpiński gasket graphs, Ars. Combin. 116 (2014) 395 405. Sierpiński triangle graphs ST n p (n N)...... are the graphs defined by: } V (STp n ) = P {s{i, j} s P ν 1, ν [n], {i, j} ( P 2 ), { } E (STp n ) = { ˆk, k n 1 {j, k}} k P, j P \ {k} { } {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}. As before, ST 0 p = K p and V (ST 0 p ) = P. S. S. Zemljič 12

Example ST 1 4 {0, 1} {0, 2} ˆ0 {0, 3} {1, 3} ˆ1 ˆ3 {2, 3} ˆ2 {1, 2} S. S. Zemljič 13

Distances in ST n p d ν (s{i, j}, ˆl) = 1 + (i = l)(j = l) + ν 1 d=1 (s d = l) 2 d [2 ν ] there are 1 + (p 2)(i = l)(j = l) s{i, j}, ˆl-shortest paths diam(st n p ) = 2 n s V (ST n p ) : p 1 d n (s, ˆl) = (p 1) 2 n l=0 d n+1 (is, jt) = min{d n (s, ĵ) + 2 n ν d ν (t, î) ; d n (s, ˆk) + 2 n ν d ν (t, ˆk) + 2 n k P \ {i, j}}. S. S. Zemljič 14

Open Problems explicit formula for average distance other metric properties which are known for Sierpiński graphs S n p we are currently working on the decision automaton for p > 3 S. S. Zemljič 15

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