On few-class Q -polynomial association schemes: feasible parameters and nonexistence results
aa r X i v : . [ m a t h . C O ] A ug On few-class Q -polynomial association schemes: feasibleparameters and nonexistence results Alexander L. Gavrilyuk ∗ Janoˇs Vidali †‡ Jason S. Williford § August 28, 2019
Abstract
We present the tables of feasible parameters of primitive 3-class Q -polynomial associationschemes and 4- and 5-class Q -bipartite association schemes (on up to 2800, 10000, and 50000 ver-tices, respectively), accompanied by a number of nonexistence results for such schemes obtainedby analysing triple intersection numbers of putative open cases. Much attention in literature on association schemes has been paid to distance-regular graphs, inparticular to those of diameter 2, also known as strongly regular graphs – however, their completeclassification is still a widely open problem. The tables of their feasible parameters, maintainedby A. E. Brouwer [4, 5], are very helpful for the algebraic combinatorics community, in particularwhen one wants to check whether a certain example has already been proven (not) to exist, to beunique, etc. Compiling such a table can be a challenging problem, as, for example, some feasibilityconditions require calculating roots of high degree polynomials.The goal of this work is to present the tables of feasible parameters of Q -polynomial associationschemes, compiled by the third author, and accompanied by a number of nonexistence resultsobtained by the first two authors.Recall that Q -polynomial association schemes can be seen as a counterpart of distance-regulargraphs, which, however, remains much less explored, although they have received considerableattention in the last few years [10, 22, 24, 25] due to their connection with some objects in quantuminformation theory such as equiangular lines and real mutually unbiased bases [21].More precisely, let A , . . . , A D and E , . . . , E D denote the adjacency matrices and the primitiveidempotents of an association scheme, respectively. An association scheme is P -polynomial (or metric ) if, after suitably reordering the relations, there exist polynomials v i of degree i such that A i = v i ( A ) (0 ≤ i ≤ D ). If this is the case, the matrix A i can be seen as the distance- i adjacency ∗ Center for Math Research and Education, Pusan National University, 2, Busandaehak-ro 63beon-gil, Geumjeong-gu, Busan, 46241, Republic of Korea. [email protected] † Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 21, 1000 Ljubljana, Slovenia. ‡ Institute of Mathematics, Physics and Mechanics, Jadranska ulica 19, 1000 Ljubljana, Slovenia. [email protected] § Department of Mathematics and Statistics, University of Wyoming, 1000 E. University Ave., Laramie, WY 82071,United States of America. [email protected] Q -polynomial(or cometric ) if, after suitably reordering the eigenspaces, there exist polynomials v ∗ j of degree j suchthat E j = v ∗ j ( E ) (0 ≤ j ≤ D ), where the matrix multiplication is entrywise. These notions are dueto Delsarte [13], who introduced the P -polynomial property as an algebraic definition of associationschemes generated by distance-regular graphs, and then defined Q -polynomial association schemesas the dual concept to P -polynomial association schemes.Many important examples of P -polynomial association schemes, which arise from classical al-gebraic objects such as dual polar spaces and forms over finite fields, also possess the Q -polynomialproperty. Bannai and Ito [1] posed the following conjecture: Conjecture 1.1.
For D large enough, a primitive association scheme of D classes is P -polynomialif and only if it is Q -polynomial. We are not aware of any progress towards its proof. The discovery of a feasible set of parametersof counter-examples (see [27]) casts some doubt on the conjecture, and in the very least shows thatthis will likely be difficult to prove. Moreover, the problem of classification of association schemeswhich are both P - and Q -polynomial (i.e., Q -polynomial distance-regular graphs) is still open. Werefer the reader to [12] for its current state.Recall that, for a P -polynomial association scheme defined on a set X , its intersection numbers p kij satisfy the triangle inequality : p kij = 0 if | i − j | > k or i + j < k , which naturally gives rise to agraph structure on X . Perhaps, due to the lack of such an intuitive combinatorial characterization,much less is known about Q -polynomial association schemes when the P -polynomial property isabsent (which also indicates that there should be much more left to discover). To date, only fewexamples of Q -polynomial schemes are known which are neither P -polynomial nor duals of P -polynomial schemes [25] – most of them are imprimitive and related to combinatorial designs. Thefirst infinite family of primitive Q -polynomial schemes that are not also P -polynomial was recentlyconstructed in [28]. Due to Conjecture 1.1, it seems that the most promising area for constructingnew examples of Q -polynomial association schemes which are not P -polynomial includes those withfew classes, say, in the range 3 ≤ D ≤
6. The tables of feasible parameters of primitive 3-class Q -polynomial association schemes and 4- and 5-class Q -bipartite association schemes presented inSection 3 may serve as a source for new constructions.The parameters of P -polynomial association schemes are restricted by a number of conditionsimplied by the triangle inequality. On the other hand, the Q -polynomial property allows us toconsider triple intersection numbers with respect to some triples of vertices, which can be thoughtof as a generalization of intersection numbers to triples of starting vertices instead of pairs. Thistechnique has been previously used by various researchers [7, 9, 15, 18, 19, 20, 33, 34], mostly toprove nonexistence of some strongly regular and distance-regular graphs with equality in the so-called Krein conditions, in which case combining the restrictions implied by the triangle inequalitywith triple intersection numbers seems the most fruitful. Yet, while calculating triple intersectionnumbers when the P -polynomial property is absent is harder, we managed to rule out a numberof open cases from the tables. This includes a putative Q -polynomial association scheme on 91vertices whose existence has been open since 1999 [11].The paper is organized as follows. In Section 2, we recall the basic theory of associationschemes and their triple intersection numbers. In Section 3, we comment on the tables of feasibleparameters of Q -polynomial association schemes and how they were generated. In Section 4, weexplain in details the analysis of triple intersection numbers of Q -polynomial association schemesand prove nonexistence for many open cases from the tables. Finally, in Section 5, we discuss the2eneralization of triple intersection numbers to quadruples of vertices. In this section we prepare the notions needed in subsequent sections.
Let X be a finite set of vertices and { R , R , . . . , R D } be a set of non-empty subsets of X × X . Let A i denote the adjacency matrix of the (di-)graph ( X, R i ) (0 ≤ i ≤ D ). The pair ( X, { R i } Di =0 ) iscalled a (symmetric) association scheme of D classes (or a D -class scheme for short) if the followingconditions hold:(1) A = I | X | , which is the identity matrix of size | X | ,(2) P Di =0 A i = J | X | , which is the square all-one matrix of size | X | ,(3) A ⊤ i = A i (1 ≤ i ≤ D ),(4) A i A j = P Dk =0 p kij A k , where p kij are nonnegative integers (0 ≤ i, j ≤ D ).The nonnegative integers p kij are called intersection numbers : for a pair of vertices x, y ∈ X with( x, y ) ∈ R k and integers i , j (0 ≤ i, j, k ≤ D ), p kij equals the number of vertices z ∈ X such that( x, z ) ∈ R i , ( y, z ) ∈ R j .The vector space A over R spanned by the matrices A i forms an algebra. Since A is com-mutative and semisimple, there exists a unique basis of A consisting of primitive idempotents E = | X | J | X | , E , . . . , E D (i.e., projectors onto the common eigenspaces of A , . . . , A D ). Since thealgebra A is closed under the entry-wise multiplication denoted by ◦ , we define the Krein parameters q kij (0 ≤ i, j, k ≤ D ) by E i ◦ E j = 1 | X | D X k =0 q kij E k . (2.1)It is known that the Krein parameters are nonnegative real numbers (see [13, Lemma 2.4]). Sinceboth { A , A , . . . , A D } and { E , E , . . . , E D } form bases of A , there exists matrices P = ( P ij ) Di,j =0 and Q = ( Q ij ) Di,j =0 defined by A i = D X j =0 P ji E j and E i = 1 | X | D X j =0 Q ji A j . (2.2)The matrices P and Q are called the first and second eigenmatrix of ( X, { R i } Di =0 ).Let n i , 0 ≤ i ≤ D , denote the valency of the graph ( X, R i ), and m j , 0 ≤ j ≤ D , denotethe multiplicity of the eigenspace of A , . . . , A D corresponding to E j . Note that n i = p ii , while m j = q jj .For an association scheme ( X, { R i } Di =0 ), an ordering of A , . . . , A D such that for each i (0 ≤ i ≤ D ), there exists a polynomial v i ( x ) of degree i with P ji = v i ( P j ) (0 ≤ j ≤ D ), is called a P -polynomial ordering of relations. An association scheme is said to be P -polynomial if it admits3 P -polynomial ordering of relations. The notion of an association scheme together with a P -polynomial ordering of relations is equivalent to the notion of a distance-regular graph – such agraph has adjacency matrix A , and A i (0 ≤ i ≤ D ) is the adjacency matrix of its distance- i graph (i.e., ( x, y ) ∈ R i precisely when x and y are at distance i in the graph), and the numberof classes equals the diameter of the graph. It is also known that an ordering of relations is P -polynomial if and only if the matrix of intersection numbers L , where L i := ( p kij ) Dk,j =0 (0 ≤ i ≤ D ),is a tridiagonal matrix with nonzero superdiagonal and subdiagonal [1, p. 189] – then p kij = 0holds whenever the triple ( i, j, k ) does not satisfy the triangle inequality (i.e., when | i − j | < k or i + j > k ). For a P -polynomial ordering of relations of an association scheme, set a i = p i ,i , b i = p i ,i +1 , and c i = p i ,i − . These intersection numbers are usually gathered in the intersectionarray { b , b , . . . , b D − ; c , c , . . . , c D } , as the remaining intersection numbers can be computed fromthem (in particular, a i = b − b i − c i for all i , where b D = c = 0). For an association schemewith a P -polynomial ordering of relations, the ordering E , . . . , E D is called the natural ordering of eigenspaces if ( P i ) Di =0 is a decreasing sequence.Dually, for an association scheme ( X, { R i } Di =0 ), an ordering of E , . . . , E D such that for each i (0 ≤ i ≤ D ), there exists a polynomial v ∗ i ( x ) of degree i with Q ji = v ∗ i ( Q j ) (0 ≤ j ≤ D ), iscalled a Q -polynomial ordering of eigenspaces. An association scheme is said to be Q -polynomial if it admits a Q -polynomial ordering of eigenspaces. Similarly as before, it is known that anordering of eigenspaces is Q -polynomial if and only if the matrix of Krein parameters L ∗ , where L ∗ i := ( q kij ) Dk,j =0 (0 ≤ i ≤ D ), is a tridiagonal matrix with nonzero superdiagonal and subdiagonal [1,p. 193] – then q kij = 0 holds whenever the triple ( i, j, k ) does not satisfy the triangle inequality.For a Q -polynomial ordering of eigenspaces, set a ∗ i = q i ,i , b ∗ i = q i ,i +1 , and c ∗ i = q i ,i − . Again,these Krein parameters are usually gathered in the Krein array { b ∗ , b ∗ , . . . , b ∗ D − ; c ∗ , c ∗ , . . . , c ∗ D } containing all the information needed to compute the remaining Krein parameters (in particular,we have a ∗ i = b ∗ − b ∗ i − c ∗ i for all i , where b ∗ D = c ∗ = 0). For an association scheme with a Q -polynomial ordering of eigenspaces, the ordering A , . . . , A D is called the natural ordering ofrelations if ( Q i ) Di =0 is a decreasing sequence. Unlike for the P -polynomial association schemes,there is no known general combinatorial characterization of Q -polynomial association schemes.An association scheme is called primitive if all of A , . . . , A D are adjacency matrices of connectedgraphs. It is known that a distance-regular graph is imprimitive precisely when it is a cycle ofcomposite length, an antipodal graph, or a bipartite graph (possibly more than one of these),see [5, Thm. 4.2.1]. The last two properties can be recognised from the intersection array as b i = c D − i (0 ≤ i ≤ D , i = ⌊ D/ ⌋ ) and a i = 0 (0 ≤ i ≤ D ), respectively. We may define dualproperties for a Q -polynomial association scheme – we say that it is Q -antipodal if b ∗ i = c ∗ D − i (0 ≤ i ≤ D , i = ⌊ D/ ⌋ ), and Q -bipartite if a ∗ i = 0 (0 ≤ i ≤ D ). All imprimitive Q -polynomialassociation schemes are schemes of cycles of composite length, Q -antipodal or Q -bipartite (again,possibly more than one of these). The original classification theorem by Suzuki [31] allowed twomore cases, which have however been ruled out later [8, 32]. An association scheme that is both P - and Q -polynomial is Q -antipodal if and only if it is bipartite, and is Q -bipartite if and only ifit is antipodal.A formal dual of an association scheme with first and second eigenmatrices P and Q is anassociation scheme such that, for some orderings of its relations and eigenspaces, its first and secondeigenmatrices are Q and P , respectively. Note that this duality occurs on the level of parameters –an association scheme might have several formal duals, or none at all (we can speak of duality whenthere exists a regular abelian group of automorphisms, see [5, § = Q for some orderings of its relations and eigenspaces is called formally self-dual . For suchorderings, p kij = q kij (0 ≤ i, j, k ≤ D ) holds – in particular, a formally self-dual association schemeis P -polynomial if and only if it is Q -polynomial, and then its intersection array matches its Kreinarray.Any primitive association scheme with two classes is both P - and Q -polynomial for either ofthe two orderings of relations and eigenspaces. The graph with adjacency matrix A of such ascheme is said to be strongly regular (an SRG for short) with parameters ( n, k, λ, µ ), where n = | X | is the number of vertices, k = p is the valency of each vertex, and each two distinct vertices haveprecisely λ = p common neighbours if they are adjacent, and µ = p common neighbours ifthey are not adjacent. In the sequel, we will identify P -polynomial association schemes with theircorresponding strongly regular or distance-regular graphs.By a parameter set of an association scheme, we mean the full set of p kij , q kij , P ij and Q ij describedin this section, which are real numbers satisfying the identities in [5, Lemma 2.2.1, Lemma 2.3.1].We say that a parameter set for an association scheme is feasible if it passes all known condition forthe existence of a corresponding association scheme. For distance-regular graphs, there are manyknown feasibility conditions, see [5, 12, 34]. For Q -polynomial association schemes, much less isknown – see Section 3 for the feasibility conditions we have used. For a triple of vertices x, y, z ∈ X and integers i , j , k (0 ≤ i, j, k ≤ D ) we denote by h x y zi j k i (orsimply [ i j k ] when it is clear which triple ( x, y, z ) we have in mind) the number of vertices w ∈ X such that ( x, w ) ∈ R i , ( y, w ) ∈ R j and ( z, w ) ∈ R k . We call these numbers triple intersectionnumbers .Unlike the intersection numbers, the triple intersection numbers depend, in general, on theparticular choice of ( x, y, z ). Nevertheless, for a fixed triple ( x, y, z ), we may write down a systemof 3 D linear Diophantine equations with D triple intersection numbers as variables, thus relatingthem to the intersection numbers, cf. [19]: D X ℓ =0 [ ℓ j k ] = p tjk , D X ℓ =0 [ i ℓ k ] = p sik , D X ℓ =0 [ i j ℓ ] = p rij , (2.3) where ( x, y ) ∈ R r , ( x, z ) ∈ R s , ( y, z ) ∈ R t , and[0 j k ] = δ jr δ ks , [ i k ] = δ ir δ kt , [ i j
0] = δ is δ jt . Moreover, the following theorem sometimes gives additional equations.
Theorem 2.1. ([9, Theorem 3], cf. [5, Theorem 2.3.2])
Let ( X, { R i } Di =0 ) be an association schemeof D classes with second eigenmatrix Q and Krein parameters q trs (0 ≤ r, s, t ≤ D ) . Then, q trs = 0 ⇐⇒ D X i,j,k =0 Q ir Q js Q kt h x y zi j k i = 0 for all x, y, z ∈ X. Note that in a Q -polynomial association scheme, many Krein parameters are zero, and we canuse Theorem 2.1 to obtain an equation for each of them.5 Tables of feasible parameters for Q -polynomial association schemes In this section we will describe the tables of feasible parameter sets for primitive 3-class Q -polynomial schemes and 4- and 5-class Q -bipartite schemes.These tables were all completed using the MAGMA programming language (see [2]). Anyparameter set meeting the following conditions was included in the table:(1) The parameters satisfy the Q -polynomial condition.(2) All p kij are nonnegative integers, all valencies p jj are positive.(3) For each j > np jj is even (the handshaking lemma applied to the graph ( X, R j )).(4) For each j, k > p jj p jjk is even (the handshaking lemma applied to the subconstituent( { y ∈ X | ( x, y ) ∈ R j } , R k ), x ∈ X ).(5) For each j > np jj ( p jj −
1) is divisible by 6 (the number of triangles in each graph(
X, R j ) is integral).(6) All q kij are nonnegative and for each j the multiplicity q jj (i.e., the dimension of the E j -eigenspace) is a positive integer (see [5, Proposition 2.2.2]).(7) For all i, j we have P q kij =0 m k ≤ m i m j if i = j and P q kii =0 m k ≤ m i ( m i − (the absolute bound,see [5, Theorem 2.3.3] and the references therein).(8) The splitting field is at most a degree 2 extension of the rationals (see [26]).We note that there are many other conditions known for the special case of distance-regulargraphs. It was decided to apply these conditions after the construction of the table, and those notmeeting these extra conditions were labelled as nonexistent with a note as to the condition notmet. We leave as an open question whether if any of these conditions could be generalized to anycases beyond distance-regular graphs; this (perhaps faint) hope is the main reason that they areincluded in the table.We begin with the tables for Q -bipartite schemes, since this case is somewhat simpler thanthe primitive case. Schemes which are Q -bipartite are formally dual to bipartite distance-regulargraphs. As a consequence, the formal dual to [5, Theorem 4.2.2(i)] gives the Krein array forthe quotient scheme of a Q -bipartite scheme (see [24]). Namely, if the scheme has Krein array { b ∗ , b ∗ , . . . , b ∗ D − ; c ∗ , . . . , c ∗ D } and q = µ ∗ , then the Krein array of the quotient is: (cid:26) b ∗ b ∗ µ ∗ , b ∗ b ∗ µ ∗ , . . . , b ∗ m − b ∗ m − µ ∗ ; c ∗ c ∗ µ ∗ , c ∗ c ∗ µ ∗ , . . . , c ∗ m − c ∗ m µ ∗ (cid:27) , where m = ⌊ D ⌋ . When D = 4 , m = 2, so the quotient structure is a strongly regulargraph. A database of strongly regular graph parameters up to 5000 vertices can be generatedvery quickly. From there, we note that the quotient scheme has multiplicities 1 , m , m , and that m + m = 1 + m + m for 4-class and m + m + m = 1 + m + m for 5-class schemes. Using6he identities of [5, Lemma 2.3.1], it is easily seen the Krein arrays for Q -bipartite 4- and 5-classassociation schemes are (cid:26) m , m − , m ( m − m + 1) m , m ( m − m + m − m ;1 , m ( m − m , m ( m − m + 1) m , m (cid:27) and (cid:26) m , m − , m ( m − m +1) m , m ( m − m + m − m , m ( m − m + m − m +1) m ;1 , m ( m − m , m ( m − m +1) m , m ( m − m + m − m , m (cid:27) From this it is clear that the multiplicities determine all the parameters of the scheme.In the 4-class case, the parameters are entirely determined by the quotient’s multiplicities (witha chosen Q -polynomial ordering) and m . To search, we take a strongly regular graph parameterset, choose one of two possible orderings for its multiplicities, calling its multiplicities m = 1, m , m . From the absolute bound, we have 1 + m ≤ m ( m +1)2 , and from the positivity of c ∗ we have ( m − m +1) m m ≥
0. We then search over all p m ) − ≤ m ≤ m , checking the conditionsabove. Given that we are iterating over SRG parameters together with two orderings and oneinteger, this search is very fast. The limitation of the table to 10000 vertices is mainly readabilityand practicality. The third author has unpublished tables (without comments or details) to 100000vertices, and could probably go much further without trouble.We note that Q -bipartite schemes with 5-classes are very similar, except we must iterate overboth m and m . Again, this is a very quick search, but the relative scarcity of 5-class parameter setsmakes listing up to 50000 vertices, with annotation, manageable. The table actually goes slightlyhigher, to 50520 vertices, because of the existence of an example on that number of vertices.The trickiest search was the primitive 3-class Q -polynomial parameter sets. In this case, thereis no non-trivial quotient scheme to build on.We use the following observation. Theorem 3.1.
A primitive Q -polynomial association scheme of classes must have a matrix L i with distinct eigenvalues.Proof. Assume not. If a matrix A i has only two distinct eigenvalues, it is either complete, contra-dicting the fact that it is a 3-class scheme, or a disjoint union of more than one complete graph,contradicting the fact the scheme is primitive. Therefore, the only case left to consider is when A , A , A all have three distinct eigenvalues. (We note in passing that this implies that the corre-sponding graphs are all strongly regular. In this case, the scheme would be called “amorphic”, formore on amorphic schemes see [17]).After reordering the eigenspaces and noting that the P -matrix is nonsingular, we are left withthe following for P : n n n a c e a d f b c f , n = 1 + n + n + n , and d = b + c − a , e = − − a − c . Solving for Q = nP − we obtain: n +( n − fa − b n +( n − ca − b n +( n − aa − b n − fa − b n − ca − b − n + n − aa − b n − fa − b − n + n − ca − b n − aa − b − n + n − fa − b n − ca − b n − aa − b . We arrive at our final contradiction: since each column contains a repeated entry, the Q -matrixcannot be generated by one column via polynomials.We note that, in fact, all Q -polynomial D -class schemes must have a relation with D + 1 distincteigenvalues. However, the above theorem and its proof is sufficient for our needs.From this we conclude that each 3-class primitive Q -polynomial scheme has an adjacency matrix,which we label A , which has four distinct eigenvalues. Then the corresponding 4 × L has four distinct eigenvalues. From this matrix, all of the other parameters may bedetermined. In particular, from [5, Proposition 2.2.2], the left-eigenvectors of L , normalized sotheir leftmost entry is 1, must be the rows of P .The rest of the parameters can be derived from the equations: L j = P − diag( P j , P j , . . . , P Dj ) P,L ∗ j = Q − diag( Q j , Q j , . . . , Q Dj ) Q. However, checking the Q -polynomial condition is done before the computation of all parameters.We use the following theorem, a proof of which can be found in [27]. Theorem 3.2.
Let L i be an intersection matrix of a D -class association scheme, where L i hasexactly D + 1 distinct eigenvalues. Then the scheme is Q -polynomial if and only if there is aVandermonde matrix U such that U − L i U = T where T is upper triangular. It is not hard to show that without loss of generality we can take T to be 0, implying that thefirst column of U is an eigenvector of L . We only then need to iterate over the three (nontrivial)eigenvectors of L to check this condition. If the Q -polynomial condition is met, the rest of theparameters are computed and checked for the above conditions.The schemes are then split into cases, depending on whether there is a strongly regular graphas a relation, and whether the splitting field is rational or not:(1) Diameter 3 distance-regular graphs (DRG for short).(2) No diameter 3 DRG, there is a strongly regular graph as a relation, the splitting field is therational field.(3) No diameter 3 DRG, there is a strongly regular graph as a relation, the splitting field is adegree-2 extension of the rational field.(4) No diameter 3 DRG, there is no strongly regular graph as a relation, the splitting field is therational field.(5) No diameter 3 DRG, there is no strongly regular graph as a relation, the splitting field is adegree-2 extension of the rational field. 8e note that we do not have any examples of primitive, 3-class Q -polynomial schemes withan irrational splitting field, but there are open parameter sets of such. It would be interesting todetermine if these exist. We also point out that all the feasible parameter sets known to us haverational Krein parameters. Case 1.
For DRG’s, we iterated over the number of vertices, intersection array and valencies.The order was n , b = n , b , n (noting n is a divisor of n b ), then b (noting b must be amultiple of n gcd( n ,n ) , where n = n − n − n ), from which the rest could be determined.When there is no DRG, it is tempting to try to formally dualize the above process. However,the Krein parameters of a scheme do not have to be integral, or even rational. For this reason,it seemed more advantageous to iterate over parameters that needed to be integral, namely theparameters p kij . All arithmetic was done in MAGMA using the rational field, or a splitting field ofa degree two irreducible polynomial over the rationals. Floating point arithmetic was avoided tominimize numerical errors.For the rest of the cases, L and the valencies were iterated over. In particular, the parameters a = p , b = p and c = p , together with n, n , n determine the rest of L , noting that a + b ≤ n − c ≤ n − n an . Any matrix without 4 distinct eigenvalues or with an irreducible cubicfactor in its characteristic polynomial was discarded. Cases 2 and 3.
In these cases, we iterate over strongly regular graphs first, with parameters( n, k, λ, µ ). We choose A to be the adjacency matrix of the strongly regular graph relation, and L , L to be fissions of the complement. Given this, the choice of n will determine n . Thepossibilities for n can be narrowed by observing that p = µ , n = k and p n = p n , implyingthat n is divisible by n gcd( n ,µ ) .Using similar identities, we find b is divisible by n gcd( n ,n ) , a is divisible by n gcd( n ,n ) , and c = n ( n − b − µ ) n . After choosing these parameters all of L follows. Cases 4 and 5.
In these cases, we know L , L and L all have 4 distinct eigenvalues. Therefore,we can assume n is the smallest valency, and that a ≤ b . Using a is divisible by n gcd( n ,n ) , b isdivisible by n gcd( n ,n ) , and n divides an , we choose n , a, n , b, c , from which the rest is determined.This is the slowest part of the search, and the reason the primitive table goes to 2800 vertices.We close with some comments on the irrational splitting field case. The 2-class primitive Q -polynomial case is equivalent to (primitive) strongly regular graphs. The only case where stronglyregular graphs have an irrational splitting field is the so-called “half-case”, when the graph hasvalency n − . Such graphs do exist, for example the Paley graphs for non-square prime powers q with q congruent to 1 modulo 4. We note that no primitive Q -polynomial schemes with morethan 2 classes and a quadratic splitting field are known. All feasible parameter sets we know ofare 3-class and have a strongly regular graph relation (case 3). The corresponding strongly regulargraphs are also all unknown (see [4]). We have no feasible parameter set for case 5. However, onecase 5 parameter set satisfied all criteria except the handshaking lemma. It is listed below, thoughnot included in the online table. Given this, we expect feasible parameter sets for case 5 to exist,but may be quite large. P = √ − √
19 18 − √ − −
31 19 − √ − − √
19 18+9 √ , Q = √ − − √ − √ − − √ − √ −
193 8+4 √ , = , L = , L = ,L ∗ = √ − √ − √ √ ,L ∗ = − √ √ √ − √ − √ √ ,L ∗ = − √ √ √ − √ . While feasible parameters may exist, the complete lack of examples elicits the following question:
Question 3.3.
Do all -class primitive Q -polynomial schemes have a rational splitting field? This is a special case of the so-called “Sensible Caveman” conjecture of William J. Martin:
Conjecture 3.4.
For Q -polynomial schemes of or more classes, if the scheme is primitive thenits splitting field is rational. We derived our nonexistence results by analyzing triple intersection numbers of Q -polynomialassociation schemes. For some choice of relations R r , R s , R t , the system of Diophantine equationsderived from (2.3) and Theorem 2.1 may have multiple nonnegative solutions, each giving thepossible values of the triple intersection numbers with respect to a triple ( x, y, z ) with ( x, y ) ∈ R r ,( x, z ) ∈ R s and ( y, z ) ∈ R t . However, in certain cases, there might be no nonnegative solutions –in this case, we may conclude that an association scheme with the given parameters does not exist.Even when there are solutions for all choices of R r , R s , R t such that p trs = 0, sometimes nonex-istence can be derived by other means. We may, for example, employ double counting. Proposition 4.1.
Let x and y be vertices of an association scheme with ( x, y ) ∈ R r . Supposethat α , α , . . . , α m are distinct integers such that there are precisely κ ℓ vertices z with ( x, z ) ∈ R s , ( y, z ) ∈ R t and h x y zi j k i = α ℓ (1 ≤ ℓ ≤ m , P mℓ =1 κ ℓ = p rst ) , and β , β , . . . , β n are distinct integerssuch that there are precisely λ ℓ vertices w with ( w, x ) ∈ R i , ( w, y ) ∈ R j and h w x yk s t i = β ℓ (1 ≤ ℓ ≤ n , P nℓ =1 λ ℓ = p rij ) . Then, m X ℓ =1 κ ℓ α ℓ = n X ℓ =1 λ ℓ β ℓ . roof. Count the number of pairs ( w, z ) with ( x, z ) ∈ R s , ( y, z ) ∈ R t , ( w, x ) ∈ R i , ( w, y ) ∈ R j and( w, z ) ∈ R k .We consider the special case of Proposition 4.1 when a triple intersection number is zero for alltriples of vertices in some given relations. Corollary 4.2.
Suppose that for all vertices x, y, z of an association scheme with ( x, y ) ∈ R r , ( x, z ) ∈ R s , ( y, z ) ∈ R t , h x y zi j k i = 0 holds. Then, h w x yk s t i = 0 holds for all vertices w, x, y with ( w, x ) ∈ R i , ( w, y ) ∈ R j and ( x, y ) ∈ R r .Proof. Apply Proposition 4.1 to all ( x, y ) ∈ R r , with m ≤ α = 0. Since β ℓ and λ ℓ (1 ≤ ℓ ≤ n )must be nonnegative, it follows that n ≤ β = 0. The sage-drg package [35, 34] by the second author for the SageMath computer algebra system [29]has been used to perform computations of triple intersection numbers of Q -polynomial associationschemes with Krein arrays that were marked as open in the tables of feasible parameter sets by thethird author [36], see Section 3. The package was originally developed for the purposes of feasibilitychecking for intersection arrays of distance-regular graphs and included a routine to find generalsolutions to the system of equations for computing triple intersection numbers.For the purposes of the current research, the package has been extended to support parametersof general association schemes, in particular, given as Krein arrays of Q -polynomial associationschemes. Additionally, the package now supports generating integral solutions for systems of equa-tions with constraints on the solutions (e.g., nonnegativity of triple intersection numbers) – thesecan also be added on-the-fly. The routine uses SageMath’s mixed integer linear programming facil-ities, which support multiple solvers. We have used SageMath’s default GLPK solver [23] and theCBC solver [14] in our computations – however, other solvers can also be used if they are available.We have thus been able to implement an algorithm which tries to narrow down the possiblesolutions of the systems of equations for determining triple intersection numbers of an associationscheme such that they satisfy Corollary 4.2, and conclude inequality if any of the systems ofequations has no such feasible solutions.(1) For each triple of relations ( R r , R s , R t ) such that p trs >
0, initialize an empty set of solutions,obtain a general (i.e., parametric) solution to the system of equations derived from (2.3) andTheorem 2.1, and initialize a generator of solutions with the constraint that the intersectionnumbers be integral and nonnegative. All generators ( r, s, t ) are initially marked as active ,and all triple intersection numbers ( r, s, t ; i, j, k ) (representing h x y zi j k i with ( x, y ) ∈ R r ,( x, z ) ∈ R s and ( y, z ) ∈ R t ) are initially marked as unknown .(2) For each active generator, generate one solution and add it to the corresponding set of solu-tions. If a generator does not return a new solution (i.e., it has exhausted all of them), thenmark it as inactive .(3) For each inactive generator, verify that the corresponding set of solutions is nonempty –otherwise, terminate and conclude nonexistence.(4) Initialize an empty set Z . 115) For each unknown triple intersection number ( r, s, t ; i, j, k ), mark it as nonzero if a solutionhas been found in which its value is not zero. If such a solution has not been found yet, makea copy of the generator ( r, s, t ) with the constraint that ( r, s, t ; i, j, k ) be nonzero, and generateone solution. If such a solution exists, add it to the set of solutions and mark ( r, s, t ; i, j, k )as nonzero , otherwise mark ( r, s, t ; i, j, k ) as zero and add it to Z .(6) If Z is empty, terminate without concluding nonexistence.(7) For each triple intersection number ( r, s, t ; i, j, k ) ∈ Z and for each nonzero ( a, b, c ; d, e, f ) ∈{ ( r, i, j ; s, t, k ) , ( s, i, k ; r, t, j ) , ( t, j, k ; r, s, i ) } , remove all solutions from the corresponding setin which the value of the latter is nonzero, mark ( a, b, c ; d, e, f ) as zero , mark all nonzero ( a, b, c ; ℓ, m, n ) with ( ℓ, m, n ) = ( d, e, f ) as unknown , and add a constraint that ( a, b, c ; d, e, f )be zero to the generator ( a, b, c ) if it is active .(8) Go to (2).Note that generators and triple intersection numbers are considered equivalent under permuta-tion of vertices, i.e., under actions ( r, s, t ) ( r, s, t ) π and ( r, s, t ; i, j, k ) (( r, s, t ) π ; ( i, j, k ) π (1 3) )for π ∈ S .The above algorithm is available as the check quadruples method of sage-drg ’s ASParameters class. We ran it for all open cases in the tables from Section 3, and obtained 29 nonexistenceresults for primitive 3-class schemes, 92 nonexistence results for Q -bipartite 4-class schemes, and11 nonexistence results for Q -bipartite 5-class schemes. The results are summarized in the followingtheorem. Theorem 4.3. A Q -polynomial association scheme with Krein array listed in one of Tables 1, 2and 3 does not exist.Proof. In all but two cases, it suffices to observe that for some triple of relations R r , R s , R t , thesystem of equations derived from (2.3) and Theorem 2.1 has no integral nonnegative solutions –Tables 1 and 2 list the triple ( r, s, t ), while for all examples in Table 3, this is true for ( r, s, t ) =(1 , , h , i and h , i from Table 1. In the first case, the Kreinarray is { , , /
11; 1 , / , } . Such an association scheme has two Q -polynomial orderings,so we can augment the system of equations (2.3) with six equations derived from Theorem 2.1. Let w, x, y, z be vertices such that ( x, z ) , ( y, z ) ∈ R and ( w, x ) , ( w, y ) , ( x, y ) ∈ R . Since p = 22 and p = 3, such vertices must exist. We first compute the triple intersection numbers with respectto x, y, z . There are two integral nonnegative solutions, both having [3 3 1] = 0. On the otherhand, there is a single solution for the triple intersection numbers with respect to w, x, y , giving[1 1 1] = 3. However, this contradicts Corollary 4.2, so such an association scheme does not exist.In the second case, the Krein array is { , ,
25; 1 , , } . Let w, x, y, z be vertices such that( x, y ) , ( x, z ) ∈ R , ( w, y ) , ( y, z ) ∈ R and ( w, x ) ∈ R . Since p = 70 and p = 250, such verticesmust exist. There is a single solution for the triple intersection numbers with respect to x, y, z ,giving [3 2 3] = 0. On the other hand, there are four solutions for the triple intersection numberswith respect to w, x, y , from which we obtain [3 1 2] ∈ { , , , } . Again, this contradictsCorollary 4.2, so such an association scheme does not exist. This completes the proof. Remark 4.4.
The sage-drg package repository provides two Jupyter notebooks containing thecomputation details in the proofs of nonexistence of two cases from Table 1: abel Krein array DRG Nonexistence Family h , i { , , ; 1 , , } (3 , , h , i { , , ; 1 , , } (3 , ,
1; 3 , , h , i { , ,
10; 1 , , } (1 , ,
2) (4.3) h , i { , ,
10; 1 , , } (1 , , h , i { , ,
12; 1 , , } (1 , , h , i { , , ; 1 , , } (1 , , h , i { , , ; 1 , , } (1 , , h , i { , ,
15; 1 , , } (1 , , h , i { , ,
18; 1 , , } (1 , , h , i { , ,
14; 1 , , } (1 , , h , i { , , ; 1 , , } (1 , , h , i { , ,
21; 1 , , } FSD (1 , , h , i { , , ; 1 , , } (1 , , h , i { , , ; 1 , , } (1 , , h , i { , ,
25; 1 , , } FSD (1 , ,
2; 3 , , h , i { , , ; 1 , , } (1 , , h , i a { , ,
7; 1 , , } (1 , , h , i { , , ; 1 , , } (1 , , h , i { , ,
25; 1 , , } (1 , , h , i { , ,
28; 1 , , } FSD (1 , , h , i { , , ; 1 , , } { } (1 , , h , i { , , ; 1 , , } (1 , , h , i { , , ; 1 , , } (1 , , h , i { , ,
10; 1 , , } FSD (0231) (2 , , h , i { , ,
33; 1 , , } (1 , , h , i { , ,
12; 1 , , } (1 , , h , i { , ,
29; 1 , , } (1 , , h , i a { , ,
26; 1 , , } (1 , ,
2) (4.3) h , i { , ,
34; 1 , , } FSD (1 , , Table 1: Nonexistence results for feasible Krein arrays of primitive 3-class Q -polynomial associationschemes on up to 2800 vertices. For P -polynomial parameters (for the natural ordering of relations,unless otherwise indicated), the DRG column indicates whether the parameters are formally self-dual (FSD), or the intersection array is given. The Nonexistence column gives either the tripleof relation indices for which there is no solution for triple intersection numbers, or the 6-tuple ofrelation indices ( r, s, t ; i, j, k ) for which Corollary 4.2 is not satisfied. The Family column specifiesthe infinite family from Subsection 4.2 that the parameter set is part of.13 abel Krein array Nonexistence Family h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
5; 1 , , , } (1 , , h , i { , , ,
9; 1 , , , } (1 , ,
2) (4.4) h , i { , , ,
8; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
11; 1 , , , } (1 , ,
2) (4.4) h , i { , , ,
13; 1 , , , } (1 , ,
2) (4.4) h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
8; 1 , , , } (1 , , h , i { , , ,
15; 1 , , , } (1 , ,
2) (4.4) h , i { , , ,
17; 1 , , , } (1 , ,
2) (4.4) h , i { , , ,
19; 1 , , , } (1 , ,
2) (4.4) h , i { , , ,
16; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
21; 1 , , , } (1 , ,
2) (4.4) h , i { , , ,
15; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
9; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
12; 1 , , , } (1 , , h , i { , , ,
25; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
16; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
28; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
21; 1 , , , } (1 , , h , i { , , ,
24; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
12; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
35; 1 , , , } (1 , , (Continued on next page.) Continued.)
Label Krein array Nonexistence Family h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
25; 1 , , , } (1 , ,
2) (4.4) h , i { , , ,
21; 1 , , , } (1 , , h , i { , , ,
27; 1 , , , } (1 , ,
2) (4.4) h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
24; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
29; 1 , , , } (1 , ,
2) (4.4) h , i { , , ,
36; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
49; 1 , , , } (1 , , h , i a { , , ,
25; 1 , , , } (1 , , h , i { , , ,
31; 1 , , , } (1 , ,
2) (4.4) h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
45; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
35; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
33; 1 , , , } (1 , ,
2) (4.4) h , i { , , ,
35; 1 , , , } (1 , ,
2) (4.4) h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
37; 1 , , , } (1 , ,
2) (4.4) h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
39; 1 , , , } (1 , ,
2) (4.4) h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
40; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
41; 1 , , , } (1 , ,
2) (4.4) h , i { , , ,
36; 1 , , , } (1 , , h , i { , , ,
16; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
13; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
43; 1 , , , } (1 , ,
2) (4.4) h , i { , , , ; 1 , , , } (1 , , h , i { , , , ; 1 , , , } (1 , , h , i { , , ,
27; 1 , , , } (1 , , h , i { , , ,
45; 1 , , , } (1 , ,
2) (4.4) h , i { , , ,
49; 1 , , , } (1 , , h , i { , , ,
56; 1 , , , } (1 , , Table 2: Nonexistence results for feasible Krein arrays of Q -bipartite (but not Q -antipodal) 4-class Q -polynomial association schemes on up to 10000 vertices. The Nonexistence column gives eitherthe triple of relation indices for which there is no solution for triple intersection numbers. TheFamily column specifies the infinite family from Subsection 4.2 that the parameter set is part of.15 abel Krein array Family h , i { , , , , ; 1 , , , , }h , i { , , , , ; 1 , , , , } (4.5) h , i { , , , , ; 1 , , , , }h , i { , , , , ; 1 , , , , }h , i { , , , , ; 1 , , , , } (4.5) h , i { , , , , ; 1 , , , , } (4.5) h , i { , , , , ; 1 , , , , } (4.5) h , i { , , , , ; 1 , , , , } (4.5) h , i { , , , , ; 1 , , , , }h , i { , , , , ; 1 , , , , } (4.5) h , i { , , , , ; 1 , , , , } Table 3: Nonexistence results for feasible Krein arrays of Q -bipartite (but not Q -antipodal) 5-class Q -polynomial association schemes on up to 50000 vertices. In all cases, there is no solution fortriple intersection numbers for a triple of vertices mutually in relation R . The Family columnspecifies the infinite family from Subsection 4.2 that the parameter set is part of. • QPoly-24-20-36 11-1-30 11-24.ipynb for the case h , i , and • DRG-104-70-25-1-7-80.ipynb for the case h , i . Remark 4.5.
The parameter set h , i from Table 1 was listed by Van Dam [11] as the smallestfeasible Q -polynomial parameter set for which no scheme is known. The next such open case is nowthe Krein array { , / , /
4; 1 , / , / } for a primitive -class Q -polynomial associationscheme with vertices, which was also listed by Van Dam. Since some of the parameters from Table 1 also admit a P -polynomial ordering, we can derivenonexistence of distance-regular graphs with certain intersection arrays. We have also found anintersection array for a primitive Q -polynomial distance-regular graph of diameter 4, which is listedin [5] and [3], and for which, to the best of our knowledge, nonexistence has not been previouslyknown. Theorem 4.6.
There is no distance-regular graph with intersection array { , ,
21; 1 , , } , { , ,
25; 1 , , } , { , ,
28; 1 , , } , { , ,
24; 1 , , } , { , ,
10; 1 , , } , { , ,
34; 1 , , } , or { , , ,
16; 1 , , , } . Proof.
The cases h , i , h , i , h , i and h , i from Table 1 are formally self-dual for the natural ordering of relations, while h , i is formally self-dual with ordering ofrelations A , A , A relative to the natural ordering. In each case, the corresponding associationscheme is P -polynomial with intersection array equal to the Krein array. The case h , i is not formally self-dual, yet the natural ordering of relations is P -polynomial with intersection16rray { , ,
24; 1 , , } . In all of the above cases, Theorem 4.3 implies nonexistence of thecorresponding association scheme, so a distance-regular graph with such an intersection array doesnot exist.Consider now a distance-regular graph with intersection array { , , ,
16; 1 , , , } . Sucha graph is formally self-dual for the natural ordering of eigenspaces and therefore also Q -polynomial.Augmenting the system of equations (2.3) with twelve equations derived from Theorem 2.1 givesa two parameter solution for triple intersection numbers with respect to three vertices mutually atdistances 1 , ,
3. However, it turns out that there is no integral solution, leading to nonexistence ofthe graph.
Remark 4.7.
The non-existence of a distance-regular graph with intersection array { , , , , , , } also follows by applying the Terwilliger polynomial [15]. Recall that this polynomial, say T Γ ( x ) , which depends only on the intersection numbers of a Q -polynomial distance-regular graph Γ and its Q -polynomial ordering, satisfies: T Γ ( η ) ≥ , (4.1) where η is any non-principal eigenvalue of the local graph of an arbitrary vertex x of Γ . Furthermore,by [5, Theorem 4.4.3(i)], η satisfies − − b θ + 1 ≤ η ≤ − − b θ D + 1 , (4.2) where b = θ > θ > . . . > θ D are the D + 1 distinct eigenvalues of Γ .For the above-mentioned intersection array, T Γ ( x ) is a polynomial of degree with a negativeleading term and the following roots: − (= − − b θ +1 ) , −√ ≈ − . , (= − − b θ D +1 ) , √ ≈ . .Thus, combining (4.1) and (4.2) , we obtain − ≤ η ≤ − √ or η = 173 , and one can finally obtain a contradiction as in [16, Claim 4.3]. The data from Tables 1, 2 and 3 allows us to look for infinite families of Krein arrays for which wecan show nonexistence of corresponding Q -polynomial association schemes. We find three families,one for each number of classes.The first family of Krein arrays is given by { r − , r − , r + 1; 1 , , r − } . (4.3)This Krein array is feasible for all integers r ≥
2. A Q -polynomial association scheme with Kreinarray (4.3) has 3 classes and 4 r vertices. Examples exist when r is a power of 2 – they are realizedby duals of Kasami codes with minimum distance 5, see [5, § Theorem 4.8. A Q -polynomial association scheme with Krein array (4.3) and r odd does notexist. roof. Consider a Q -polynomial association scheme with Krein array (4.3). Besides the Kreinparameters failing the triangle inequality, q is also zero. Therefore, in order to compute tripleintersection numbers, the system of equations (2.3) can be augmented with four equations derivedfrom Theorem 2.1. We compute triple intersection numbers with respect to vertices x, y, z suchthat ( x, y ) , ( x, z ) ∈ R and ( y, z ) ∈ R . Since p = r ( r + 2)( r − / >
0, such vertices mustexist. We obtain a four parameter solution (see the notebook
QPoly-d3-1param-odd.ipynb on the sage-drg package repository for computation details). Then we may express[1 2 3] = − r r + [1 3 1] + 3 · [2 3 3] − [3 1 1] + 4 · [3 3 3] . Clearly, the above triple intersection number can only be integral when r is even. Therefore,we conclude that a Q -polynomial association scheme with Krein array (4.3) and r odd does notexist.The next family is a two parameter family of Krein arrays { m, m − , m ( r − /r , m − r + 1; 1 , m/r , r − , m } (4.4)This Krein array is feasible for all integers m and r such that 0 < r − ≤ m ≤ r ( r − r + 2)and m ( r + 1) is even. A Q -polynomial association scheme with Krein array (4.4) is Q -bipartitewith 4 classes and 2 m vertices. One may take the Q -bipartite quotient of such a scheme (i.e.,identify vertices in relation R ) to obtain a strongly regular graph with parameters ( n, k, λ, µ ) =( m , ( m − r , m + r ( r − , r ( r − Latin square type .There are several examples of Q -polynomial association schemes with Krein array (4.4) forsome r and m . For ( r, m ) = (2 ,
6) and ( r, m ) = (3 , E lattice and an overlattice of the Barnes-Wall lattice in R [25],respectively. For ( r, m ) = (2 ij , i (2 j +1) ), there are examples arising from duals of extended Kasamicodes [5, § i and j . In particular, the Krein array obtainedby setting i = j = 1 uniquely determines the halved 8-cube.In the case when r is a prime power and m = r , the formal dual of this parameter set (i.e.,a distance-regular graph with the corresponding intersection array) is realized by a Pasechnikgraph [6]. Theorem 4.9. A Q -polynomial association scheme with Krein array (4.4) and m odd does notexist.Proof. Consider a Q -polynomial association scheme with Krein array (4.4). Since the scheme is Q -bipartite, we have q kij = 0 whenever i + j + k is odd or the triple ( i, j, k ) does not satisfy thetriangle inequality. This allows us to augment the system of equations (2.3) with many equationsderived from Theorem 2.1. We compute triple intersection numbers with respect to vertices x, y, z such that ( x, y ) , ( x, z ) ∈ R and ( y, z ) ∈ R . Since p = r ( r − / >
0, such vertices must exist.We obtain a one parameter solution (see the notebook
QPoly-d4-LS-odd.ipynb on the sage-drg package repository for computation details) which allows us to express[1 1 3] = r + r (1 − r )2 − m . Clearly, the above triple intersection number can only be integral when m is even. Therefore,we conclude that a Q -polynomial association scheme with Krein array (4.4) and m odd does notexist. 18he last family is given by the Krein array (cid:26) r + 12 , r − , ( r + 1) r ( r + 1) , ( r − r + 1)4 r , r + 12 r ;1 , ( r − r + 1)2 r ( r + 1) , ( r + 1)( r + 1)4 r , ( r − r + 1)2 r , r + 12 (cid:27) (4.5)This Krein array is feasible for all odd r ≥
5. A Q -polynomial association scheme with Kreinarray (4.5) is Q -bipartite with 5 classes and 2( r + 1)( r + 1) vertices. One may take the Q -bipartite quotient of such a scheme to obtain a strongly regular graph with parameters ( n, k, λ, µ ) =(( r + 1)( r + 1) , r ( r + 1) , r − , r + 1) – these are precisely the parameters of collinearity graphsof generalized quadrangles GQ( r, r ). The scheme also has a second Q -polynomial ordering ofeigenspaces, namely the ordering E , E , E , E , E relative to the ordering implied by the Kreinarray. For r ≡ C ( r ) dual polar graph. Theorem 4.10. A Q -polynomial association scheme with Krein array (4.5) and r ≡ does not exist.Proof. Consider a Q -polynomial association scheme with Krein array (4.5). Since the scheme is Q -bipartite, we have q kij = 0 whenever i + j + k is odd or the triple ( i, j, k ) does not satisfy the triangleinequality. This allows us to augment the system of equations (2.3) with many equations derivedfrom Theorem 2.1. We compute triple intersection numbers with respect to vertices x, y, z that aremutually in relation R . Since p = ( r − / >
0, such vertices must exist. We obtain a singlesolution (see the notebook
QPoly-d5-1param-3mod4.ipynb on the sage-drg package repositoryfor computation details) with [1 1 1] = r − . Clearly, the above triple intersection number can only be integral when r ≡ Q -polynomial association scheme with Krein array (4.5) and r ≡ The argument of the proof of Theorem 2.1 ([5, Theorem 2.3.2]) can be further extended to s -tuples ofvertices (see Remark (iii) in [5, § quadruple intersection numbers with respect to a quadruple of vertices w, x, y, z ∈ X . For integers h, i, j, k (0 ≤ h, i, j, k ≤ D ),denote by h w x y zh i j k i (or simply [ h i j k ] when it is clear which quadruple ( w, x, y, z ) we have inmind) the number of vertices u ∈ X such that ( u, w ) ∈ R h , ( u, x ) ∈ R i , ( u, y ) ∈ R j , and ( u, z ) ∈ R k .For a fixed quadruple ( w, x, y, z ), one can obtain a system of linear Diophantine equations withquadruple intersection numbers as variables which relates them to the intersection numbers (or tothe triple intersection numbers).The following analogue of Theorem 2.1 allows us to obtain some additional equations. Theorem 5.1.
Let ( X, { R i } Di =0 ) be an association scheme of D classes with second eigenmatrix Q and Krein parameters q kij (0 ≤ i, j, k ≤ D ) . Then, for fixed p, r, s, t (0 ≤ p, r, s, t ≤ D ) , D X ℓ =0 q ℓpr q ℓst = 0 ⇐⇒ D X h,i,j,k =0 Q hp Q ir Q js Q kt h w x y zh i j k i = 0 for all w, x, y, z ∈ X . roof. Since E i is a symmetric idempotent matrix, one has X w ∈ X E i ( u, w ) E i ( v, w ) = E i ( u, v ) . (5.1)Let Σ( M ) denote the sum of all entries of a matrix M . Then, by (5.1),Σ( E p ◦ E r ◦ E s ◦ E t ) = X u,v ∈ X E p ( u, v ) E r ( u, v ) E s ( u, v ) E t ( u, v )= X w,x,y,z ∈ X X u ∈ X E p ( u, w ) E r ( u, x ) E s ( u, y ) E t ( u, z ) ! · X v ∈ X E p ( v, w ) E r ( v, x ) E s ( v, y ) E t ( v, z ) ! = X w,x,y,z ∈ X σ ( w, x, y, z ) ≥ , (5.2)where σ ( w, x, y, z ) = P u ∈ X E p ( u, w ) E r ( u, x ) E s ( u, y ) E t ( u, z ).On the other hand, by (2.1), | X | Σ( E p ◦ E r ◦ E s ◦ E t ) = | X | Tr(( E p ◦ E r ) · ( E s ◦ E t ))= Tr D X ℓ =0 q ℓpr E ℓ ! · D X ℓ =0 q ℓst E ℓ !! = D X ℓ =0 m ℓ q ℓpr q ℓst , (5.3)where m ℓ is the rank of E ℓ (i.e., the multiplicity of the corresponding eigenspace), and by (2.2), | X | Σ( E p ◦ E r ◦ E s ◦ E t ) = 1 | X | D X ℓ =0 Q ℓp Q ℓr Q ℓs Q ℓt Σ( A ℓ )= D X ℓ =0 n ℓ Q ℓp Q ℓr Q ℓs Q ℓt , (5.4)where n ℓ is the valency of ( X, R ℓ ).Since the multiplicities m ℓ are positive numbers and the Krein parameters are non-negativenumbers, by (5.2), (5.3), (5.4), we have Σ( E p ◦ E r ◦ E s ◦ E t ) = 0 if and only if q ℓpr q ℓst = 0 (with fixed p, r, s, t ) for all ℓ = 0 , . . . , D . In this case, we have σ ( w, x, y, z ) = 0 for all quadruples ( w, x, y, z ),which implies 0 = | X | σ ( w, x, y, z ) = | X | X u ∈ X E p ( u, w ) E r ( u, x ) E s ( u, y ) E t ( u, z )= D X h,i,j,k =0 Q hp Q ir Q js Q kt h w x y zh i j k i , which completes the proof. 20he condition of Theorem 5.1 is satisfied when, for example, an association scheme is Q -bipartite, i.e., q kij = 0 whenever i + j + k is odd (take p + r and s + t of different parity).Suda [30] lists several families of association schemes which are known to be triply regular , i.e.,their triple intersection numbers h x y zi j k i only depend on i, j, k and the mutual distances between x, y, z , and not on the choices of the vertices themselves: • strongly regular graphs with q = 0 (cf. [7]), • Taylor graphs (antipodal Q -bipartite schemes of 3 classes), • linked systems of symmetric designs (certain Q -antipodal schemes of 3 classes) with a ∗ = 0, • tight spherical 7-designs (certain Q -bipartite schemes of 4 classes), and • collections of real mutually unbiased bases ( Q -antipodal Q -bipartite schemes of 4 classes).Schemes belonging to the above families seem natural candidates for the computations of theirquadruple intersection numbers. However, the condition of Theorem 5.1 is never satisfied forprimitive strongly regular graphs, while for Taylor graphs the obtained equations do not give anyinformation that could not be obtained through relating the quadruple intersection numbers to thetriple intersection numbers. This was also the case for the examples of triply regular linked systemsof symmetric designs that we have checked. However, in the cases of tight spherical 7-designsand mutually unbiased bases, we do get new restrictions on quadruple intersection numbers. Sofar, we have not succeeded in using this new information for either new constructions or proofs ofnonexistence. Acknowledgements
Alexander Gavrilyuk is supported by BK21plus Center for Math Research and Education atPusan National University, and by Basic Science Research Program through the National Re-search Foundation of Korea (NRF) funded by the Ministry of Education (grant number NRF-2018R1D1A1B07047427). Janoˇs Vidali is supported by the Slovenian Research Agency (researchprogram P1-0285 and project J1-8130). Alexander Gavrilyuk and Janoˇs Vidali are also jointlysupported by the Slovenian Research Agency (Slovenia-Russia bilateral grant number BI-RU/19-20-007). Jason Williford was supported by National Science Foundation (NSF) grant DMS-1400281.
References [1] E. Bannai and T. Ito.
Algebraic combinatorics I: Association schemes . The Benja-min/Cummings Publishing Co., Inc., 1984.[2] W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system I, The user language.
J.Symbolic Comput. , 24:235–265, 1997. doi:10.1006/jsco.1996.0125 .[3] A. E. Brouwer. Parameters of distance-regular graphs, 2011. .[4] A. E. Brouwer. Strongly regular graphs, 2013. .215] A. E. Brouwer, A. M. Cohen, and A. Neumaier.
Distance-regular graphs , volume 18 of
Ergeb-nisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas(3)] . Springer-Verlag, Berlin, 1989. doi:10.1007/978-3-642-74341-2 .[6] A. E. Brouwer and D. V. Pasechnik. Two distance-regular graphs.
J. Algebraic Combin. ,36(3):403–407, 2012. doi:10.1007/s10801-011-0341-1 .[7] P. J. Cameron, J.-M. Goethals, and J. J. Seidel. Strongly regular graphs having strongly regularsubconstituents.
J. Algebra , 55(2):257–280, 1978. doi:10.1016/0021-8693(78)90220-X .[8] D. R. Cerzo and H. Suzuki. Non-existence of imprimitive Q -polynomial schemesof exceptional type with d = 4. European J. Combin. , 30(3):674–681, 2009. doi:10.1016/j.ejc.2008.07.014 .[9] K. Coolsaet and A. Juriˇsi´c. Using equality in the Krein conditions to prove nonexist-ence of certain distance-regular graphs.
J. Combin. Theory Ser. A , 115(6):1086–1095, 2008. doi:10.1016/j.jcta.2007.12.001 .[10] E. van Dam, W. Martin, and M. Muzychuk. Uniformity in association schemes and coherentconfigurations: cometric Q -antipodal schemes and linked systems. J. Combin. Theory Ser. A ,120(7):1401–1439, 2013. doi:10.1016/j.jcta.2013.04.004 .[11] E. R. van Dam. Three-class association schemes.
J. Algebraic Combin. , 10(1):69–107, 1999. doi:10.1023/A:1018628204156 .[12] E. R. van Dam, J. H. Koolen, and H. Tanaka. Distance-regular graphs.
Electron. J. Combin. ,DS:22, 2016. .[13] P. Delsarte. An algebraic approach to the association schemes of coding theory.
Philips Res.Rep. Suppl. , (10):vi+97, 1973.[14] J. Forrest, T. Ralphs, S. Vigerske, L. Hafer, B. Kristjansson, J. P. Fasano, E. Straver,M. Lubin, H. G. Santos, R. Lougee, and M. Saltzman. coin-or/Cbc : COIN-ORBranch-and-Cut MIP Solver, version 2.9.4, 2015. https://projects.coin-or.org/Cbc , doi:10.5281/zenodo.1317566 .[15] A. L. Gavrilyuk and J. H. Koolen. The Terwilliger polynomial of a Q -polynomial distance-regular graph and its application to pseudo-partition graphs. Linear Algebra Appl. , 466(1):117–140, 2015. doi:10.1016/j.laa.2014.09.048 .[16] A. L. Gavrilyuk and J. H. Koolen. A characterization of the graphs of bilinear ( d × d )-formsover F . Combinatorica , 39(2):289–321, 2019. doi:10.1007/s00493-017-3573-4 .[17] T. Ikuta, T. Ito, and A. Munemasa. On pseudo-automorphisms and fusions of an associationscheme.
European J. Combin. , 12(4):317–325, 1991. doi:10.1016/S0195-6698(13)80114-X .[18] A. Juriˇsi´c, J. Koolen, and P. Terwilliger. Tight distance-regular graphs.
J. Algebraic Combin. ,12(2):163–197, 2000. doi:10.1023/A:1026544111089 .[19] A. Juriˇsi´c and J. Vidali. Extremal 1-codes in distance-regular graphs of diameter 3.
Des.Codes Cryptogr. , 65(1–2):29–47, 2012. doi:10.1007/s10623-012-9651-0 .2220] A. Juriˇsi´c and J. Vidali. Restrictions on classical distance-regular graphs.
J. Algebraic Combin. ,46(3–4):571–588, 2017. doi:10.1007/s10801-017-0765-3 .[21] B. G. Kodalen.
Cometric Association Schemes . PhD thesis, 2019. arXiv:1905.06959 .[22] B. G. Kodalen. Linked systems of symmetric designs.
Algebr. Comb. , 2(1):119–147, 2019. doi:10.5802/alco.22 .[23] A. Makhorin. GLPK (GNU Linear Programming Kit) v4.63.p2, 2012. .[24] W. J. Martin, M. Muzychuk, and J. Williford. Imprimitive cometric associationschemes: constructions and analysis.
J. Algebraic Combin. , 25(4):399–415, 2007. doi:10.1007/s10801-006-0043-2 .[25] W. J. Martin and H. Tanaka. Commutative association schemes.
European J. Combin. ,30(6):1497–1525, 2009. doi:10.1016/j.ejc.2008.11.001 .[26] W. J. Martin and J. Williford. There are finitely many Q -polynomial association schemeswith given first multiplicity at least three. European J. Combin. , 30(3):698–704, 2009. doi:10.1016/j.ejc.2008.07.009 .[27] G. E. Moorhouse and J. Williford. Double covers of symplectic dual polar graphs.
DiscreteMath. , 339(2):571–588, 2016. doi:10.1016/j.disc.2015.09.015 .[28] T. Penttila and J. Williford. New families of Q -polynomial association schemes. J. Combin.Theory Ser. A , 118(2):502–509, 2011. doi:10.1016/j.jcta.2010.08.001 .[29] The Sage Developers.
SageMath, the Sage Mathematics Software System (Version 7.6) , 2017. .[30] S. Suda. Coherent configurations and triply regular association schemes ob-tained from spherical designs.
J. Combin. Theory Ser. A , 117(8):1178–1194, 2010. doi:10.1016/j.jcta.2010.03.016 .[31] H. Suzuki. Imprimitive Q -polynomial association schemes. J. Algebraic Combin. , 7(2):165–180,1998. doi:10.1023/A:1008660421667 .[32] H. Tanaka and R. Tanaka. Nonexistence of exceptional imprimitive Q -polynomialassociation schemes with six classes. European J. Combin. , 32(2):155–161, 2011. doi:10.1016/j.ejc.2010.09.006 .[33] M. Urlep. Triple intersection numbers of Q -polynomial distance-regular graphs. European J.Combin. , 33(6):1246–1252, 2012. doi:10.1016/j.ejc.2012.02.005 .[34] J. Vidali. Using symbolic computation to prove nonexistence ofdistance-regular graphs.
Electron. J. Combin. , 25(4):P4.21, 2018. .[35] J. Vidali. jaanos/sage-drg : sage-drg Sage package v0.9, 2019. https://github.com/jaanos/sage-drg/ , doi:10.5281/zenodo.3350856 .[36] J. S. Williford. Homepage, 2018.