Triple intersection numbers of metric and cometric association schemes Janoˇ s Vidali University of Ljubljana Faculty of Mathematics and Physics Joint work with Alexander Gavrilyuk May 30, 2018
Association schemes Introduction Bose-Mesner algebra Triple intersection numbers Krein parameters Results Metric and cometric schemes Association schemes ◮ Let X be a set of vertices and R = { R 0 = id X , R 1 , . . . , R d } a set of symmetric relations partitioning X 2 . ◮ ( X , R ) is said to be a d-class association scheme if there exist numbers p h ij (0 ≤ h , i , j ≤ d ) such that, for any x , y ∈ X , x R h y ⇒ | { z ∈ X | x R i z R j y } | = p h ij ◮ We call the numbers p h ij (0 ≤ h , i , j ≤ d ) intersection numbers . ◮ Problem : Does an association scheme with given parameters exist? If so, is it unique? Can we determine all such schemes?
Association schemes Introduction Bose-Mesner algebra Triple intersection numbers Krein parameters Results Metric and cometric schemes Examples ◮ Hamming schemes: X = Z d n , x R i y ⇔ weight( x − y ) = i ; ◮ Johnson schemes: X = { S ⊆ Z n | | S | = d } (2 d ≤ n ), x R i y ⇔ | x ∩ y | = d − i ; ◮ Two schemes with the same parameters:
Association schemes Introduction Bose-Mesner algebra Triple intersection numbers Krein parameters Results Metric and cometric schemes Bose-Mesner algebra ◮ Let A 0 , A 1 , . . . A d be binary matrices indexed by X with ( A i ) xy = 1 iff x R i y . ◮ These matrices can be diagonalized simultaneously and they share d + 1 eigenspaces. ◮ The matrices { A 0 , A 1 , . . . A d } are the basis of the Bose-Mesner algebra M , which has a second basis { E 0 , E 1 , . . . E d } of minimal idempotents for each eigenspace. ◮ Let P be a ( d + 1) × ( d + 1) matrix with P ij being the eigenvalue of A j corresponding to the i -th eigenspace. ◮ Let Q be such that PQ = | X | I . ◮ We call P the eigenmatrix , and Q the dual eigenmatrix .
Association schemes Introduction Bose-Mesner algebra Triple intersection numbers Krein parameters Results Metric and cometric schemes Krein parameters ◮ In the Bose-Mesner algebra M , the following relations are satisfied: d d 1 � � A j = P ij E i and E j = Q ij A i . | X | i =0 i =0 ◮ We also have d d 1 � � p h q h A i A j = ij A h and E i ◦ E j = ij E h , | X | h =0 h =0 where ◦ is the entrywise matrix product. ◮ The numbers q h ij are called the Krein parameters and are nonnegative algebraic real numbers.
Association schemes Introduction Bose-Mesner algebra Triple intersection numbers Krein parameters Results Metric and cometric schemes Metric schemes – distance-regular graphs ◮ If an association scheme satisfies p h ij � = 0 ⇒ | i − j | ≤ h ≤ i + j for some ordering of its relations, then it is said to be metric or P-polynomial . ◮ Metric association schemes correspond to distance-regular graphs, with x R i y ⇔ ∂ ( x , y ) = i . ◮ The parameters of a metric association scheme can be determined from the intersection array { k , b 1 , . . . , b d − 1 ; 1 , c 2 , . . . , c d } , where a i := p i 1 , i , b i := p i 1 , i +1 , c i := p i 1 , i − 1 and k := b 0 = a i + b i + c i (0 ≤ i ≤ d ).
Association schemes Introduction Bose-Mesner algebra Triple intersection numbers Krein parameters Results Metric and cometric schemes Examples ◮ Hamming graphs: b i = ( d − i )( n − 1), c i = i ; ◮ Johnson graphs: b i = ( d − i )( n − d − i ), c i = i 2 ; q ) 2 × { + , −} , ◮ Pasechnik graphs: X = ( F 3 ( x , u , σ ) ∼ ( y , v , τ ) ⇔ σ � = τ ∧ x − y = u × v , intersection array { q 3 , q 3 − 1 , q 3 − q , q 3 − q 2 + 1; 1 , q , q 2 − 1 , q 3 } ; ◮ Coset graphs of Kasami codes: { 2 2 t +1 − 1 , 2 2 t +1 − 2 , 2 2 t + 1; 1 , 2 , 2 2 t − 1 } .
Association schemes Introduction Bose-Mesner algebra Triple intersection numbers Krein parameters Results Metric and cometric schemes Cometric schemes ◮ If an association scheme satisfies q h ij � = 0 ⇒ | i − j | ≤ h ≤ i + j for some ordering of its eigenspaces, then it is said to be cometric or Q-polynomial . ◮ The parameters of a cometric association scheme can be determined from the Krein array { m , f 1 , . . . , f d − 1 ; 1 , g 2 , . . . , g d } , where e i := q i 1 , i , f i := q i 1 , i +1 , g i := q i 1 , i − 1 and m := f 0 = e i + f i + g i (0 ≤ i ≤ d ).
Association schemes Introduction Bose-Mesner algebra Triple intersection numbers Krein parameters Results Metric and cometric schemes Examples ◮ Hamming schemes: f i = ( d − i )( n − 1), g i = i ; ◮ Johnson schemes: f i = ( d − i ) n ( n − 1)( n + 1 − i )( n − d − i ) d ( n − d )( n + 1 − 2 i )( n − 2 i ) g i = i ( d + 1 − i ) n ( n − 1)( n − d + 1 − i ) d ( n − d )( n + 2 − 2 i )( n + 1 − 2 i ) ◮ Real mutually unbiased bases: X = � w i =1 ( B i ∪ − B i ), where B 1 , B 2 , . . . , B w are orthornormal bases of R d √ with � x , y � = ± 1 / d for all x ∈ B i , y ∈ B j with i � = j ; √ √ x R i y ⇔ � x , y � = r i , r 4 i =0 = [1 , 1 / d , 0 , − 1 / d , − 1]; Krein array { d , d − 1 , d ( w − 1) , 1; 1 , d w , d − 1 , d } ; w ◮ Codewords of dual Kasami codes: { 2 2 t +1 − 1 , 2 2 t +1 − 2 , 2 2 t + 1; 1 , 2 , 2 2 t − 1 } .
Introduction Computation Triple intersection numbers Krein condition Results Triple intersection numbers ◮ In an association scheme, the intersection numbers p h ij only depend on h , i , j . ◮ Let x , y , z ∈ X with x R W y , x R V z and y R U z . ◮ We define triple intersection numbers as � x y z � := | { w ∈ X | w R h x , w R i y , w R j z } | . h i j � x y z � may depend on the particular choice of x , y , z ! ◮ h i j � x y z � ◮ When x , y , z are fixed, we abbreviate as [ h i j ]. h i j
Introduction Computation Triple intersection numbers Krein condition Results Computing triple intersection numbers ◮ We have 3 d 2 equations connecting triple intersection numbers to p h ij : p 3 p 3 p 3 p 3 1 00 10 20 30 d p 3 p p 3 p 3 3 k 1 01 � 11 21 31 [ ℓ i j ] = p U ij − δ iW δ jV , p 3 k p 3 p 3 p 3 k 2 02 p 3 12 22 32 p k 2 3 p 3 p 3 p 3 30 k 3 03 p 3 13 23 33 ℓ =1 k 3 n 31 1 k k 2 k 3 p 3 p 3 d 20 n 32 p 3 k k 2 k 3 n p 3 � [ h ℓ j ] = p V 21 1 p 3 hj − δ hW δ jU , 33 p 3 10 k 3 22 p 3 p 3 p 3 p 3 p 3 k 3 p 3 p 3 11 ℓ =1 30 31 32 33 23 p 3 00 k 2 12 p 3 d p 3 p 3 p 3 p 3 k 2 p 3 01 20 21 22 23 13 p 3 � [ h i ℓ ] = p W hi − δ hV δ iU . 02 k p 3 p p 3 p 3 p 3 3 k 10 11 12 13 03 ℓ =1 1 p p 3 p 3 p 3 3 1 00 01 02 03 ◮ All triple intersection numbers are nonnegative integers.
Introduction Computation Triple intersection numbers Krein condition Results Krein condition ◮ Theorem ([BCN89, Theorem 2.3.2], [CJ08, Theorem 3]): Let ( X , R ) be a d -class association scheme, Q its dual eigenmatrix, and q h ij its Krein parameters. ◮ q h ij = 0 iff for all triples x , y , z ∈ X : � x y z d � � Q rh Q si Q tj = 0 r s t r , s , t =0 ◮ This gives a new equation in terms of triple intersection numbers. ◮ The sage-drg [Vid18] Sage package can use all of the above to determine the possible triple intersection numbers.
Introduction Tables of feasible parameters Triple intersection numbers Nonexistence results Results Infinite families Tables of feasible parameters ◮ Various lists of feasible intersection arrays for distance-regular graphs have been published [BCN89, BCN94, Bro11]. ◮ Recently, Williford [Wil17] has published lists of feasible Krein arrays of Q -polynomial association schemes: ◮ 3-class primitive Q -polynomial association schemes on up to 2800 vertices: 62 known examples, 359 open cases; ◮ 4-class Q -bipartite, but not Q -antipodal, Q -polynomial association schemes on up to 10000 vertices: 19 known examples, 488 open cases; and ◮ 5-class Q -bipartite, but not Q -antipodal, Q -polynomial association schemes on up to 50000 vertices: 7 known examples, 16 open cases.
Introduction Tables of feasible parameters Triple intersection numbers Nonexistence results Results Infinite families Nonexistence results ◮ We have used integer linear programming to find possible triple intersection numbers for the open cases in the lists. ◮ We have been able to prove nonexistence for ◮ 31 cases of 3-class Q -polynomial association schemes, of which 8 correspond to distance-regular graphs, ◮ 92 cases of 4-class Q -polynomial association schemes, ◮ 12 cases of 5-class Q -polynomial association schemes, of which one corresponds to a distance-regular graph, and ◮ one case of a diameter 3 non- Q -polynomial distance-regular graph.
Introduction Tables of feasible parameters Triple intersection numbers Nonexistence results Results Infinite families Closed cases of Q -polynomial association schemes ◮ Smallest closed case: { 12 , 338 / 35 , 39 / 25; 1 , 312 / 175 , 39 / 5 } on 91 vertices. ◮ Double counting has been used to settle the cases ◮ { 24 , 20 , 36 / 11; 1 , 30 / 11 , 24 } on 225 vertices, ◮ { 104 , 70 , 25; 1 , 7 , 80 } on 1470 vertices, and ◮ { 132 , 343 / 3 , 56 , 28 / 3 , 1; 1 , 28 / 3 , 56 , 343 / 3 , 132 } on 3500 vertices. ◮ Most remaining cases have been ruled out because there was no integral nonnegative solution for triple intersection numbers corresponding to a triple of vertices at some given relations. ◮ Next remaining open case: { 14 , 108 / 11 , 15 / 4; 1 , 24 / 11 , 45 / 4 } on 99 vertices.
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