Conditions for the parameters of the block graph of quasi-symmetric designs ∗

A quasi-symmetric design (QSD) is a 2-(v, k, λ) design with intersection numbers x and y with x < y. The block graph of such a design is formed on its blocks with two distinct blocks being adjacent if they intersect in y points. It is well known that the block graph of a QSD is a strongly regular graph (SRG) with parameters (b, a, c, d) with smallest eigenvalue −m = −k−x y−x . The classification result of SRGs with smallest eigenvalue −m, is used to prove that for a fixed pair (λ > 2,m > 2), there are only finitely many QSDs. This gives partial support towards Marshall Hall Jr.’s conjecture, that for a fixed λ > 2, there exist finitely many symmetric (v, k, λ)-designs. We classify QSDs with m = 2 and characterize QSDs whose block graph is the complete multipartite graph with s classes of size 3. We rule out the possibility of a QSD whose block graph is the Latin square graph LSm(n) or complement of LSm(n), for m = 3, 4. SRGs with no triangles have long been studied and are of current research interest. The characterization of QSDs with triangle-free block graph for x = 1 and y = x+1 is obtained and the non-existence of such designs with x = 0 or λ > 2(x+2) or if it is a 3-design is proven. The computer algebra system Mathematica is used to find parameters of QSDs with triangle-free block graph for 2 6 m 6 100. We also give the parameters of QSDs whose block graph parameters are (b, a, c, d) listed in Brouwer’s table of SRGs.


Introduction
A 2-(v, k, λ) design D (with b blocks and r blocks through a given point) is called quasisymmetric if the sizes of the intersection of two distinct blocks take only two values x and y, with (0 x < y < k).We can create a graph Γ = Γ(D), called the block graph of D, by joining two distinct blocks if they have the larger intersection number y.This graph is a strongly regular graph with parameters (b, a, c, d).Here, as is customary, b denotes the number of vertices of Γ, a the degree of Γ, c (resp.d), the number of vertices both adjacent (resp.non-adjacent) to any two adjacent (resp.non-adjacent) vertices.In addition, to avoid trivial cases, we assume that a strongly regular graph is neither the trivial graph nor the complete graph.Furthermore, the adjacency matrix of Γ has smallest eigenvalue −m, where m = k−x y−x .From now on, we use QSD(resp.SRG) to denote a quasi-symmetric design (resp.strongly regular graph).
It is possible to create a series of equations among the parameters of the QSD.In addition, we also have conditions on the parameters of the associated SRG.These conditions are summarized in lemmas 1 through 5 (in section 2).Lemma 5 expresses these conditions involving thirteen variables a, b, c, d, m, n, v, r, k, λ, x, y, z, where z = y − x, and n = r−λ y−x .Since the parameter m is an integer involving the design parameters, we can take advantage of the fact that m 2 and the results of Seidel [32] and Neumaier [23] for SRGs with least eigenvalue −m.This gives nine equations (Lemma 5), which are necessary conditions for the existence of a QSD.Mathematica is used in section 2 to rewrite these conditions into a sequence of necessary conditions for the existence of a QSD.
In this paper, we consider mostly proper QSDs, i.e. designs in which both the intersection numbers occur.If this condition is relaxed, and we allow at most two intersection numbers, then the family of proper and improper QSDs will include symmetric designs.Finite affine planes provide an infinite class of proper QSDs with λ = 1, while finite projective planes give an infinite class of improper QSDs.So if λ = 1, there are infinitely many proper or improper QSDs.There is a famous open conjecture of Marshall Hall, Jr. [15], Hall's Conjecture: For a fixed λ 2, there are only finitely many symmetric designs (i.e.improper QSDs), whose 'λ-value' is the given λ.For a fixed λ 2, let G λ denote the class of all QSDs, proper or improper(i.e.symmetric designs).In a personal communication to the authors of [30], N.M.Singhi [40] made the following conjecture, Singhi's Conjecture: For a fixed λ 2, G λ is finite.In ( [30], Corollary 4.2), the equivalence of Hall's and Singhi's conjectures was shown.In partial support of these conjectures, we prove in section 3, that for a fixed pair (λ 2, m 2) there are only finitely many QSDs.
From a QSD D, we get the associated SRG Γ(D).So a natural question is, given a SRG with the right parameters to be the block graph of a QSD, when is it in fact, the block graph of a QSD?In section 4, we prove some results concerning this question.The papers [39], [13], [14], and [11] contain some results on this problem.The first of these papers is phrased in the language of two class partially balanced designs (whose duals are SRGs) and uses Hasse-Minkowski theory, while the paper [11] relies on the equivalent theory of quadratic forms.According to [11]: "The question which strongly regular graphs are block graphs of quasi-symmetric designs is a difficult one and there is no chance for a the electronic journal of combinatorics 22(1) (2015), #P1.36 general answer".
In ( [13], Theorem 3.5), the authors ruled out the possibility of a QSD whose block graph is the Latin square graph LS 2 (n) or its complement.We extend this result to QSDs whose block graph is the Latin square graph LS m (n) or complement of LS m (n), for m = 3 and 4.
In section 5, we consider QSDs, whose block graph, and not the complement of its block graph is triangle-free, as was assumed in [1].The block graph Γ of a QSD D is triangle-free if and only if Γ has no co-cliques of size three.We rule out the possibility of a QSD whose block graph is triangle-free under each of the following conditions: x = 0; λ > 2(x + 2); or if it is a 3-design, or (k − x)/(y − x) = 3, 4 and 5.As a consequence, there does not exist a QSD with x = 0 and having a triangle-free block graph.This shows that the analogue of the conjecture in [20] (by requiring Γ, and not Γ to be triangle-free) is true.We also characterize QSDs with k = 2y − x or y = x + 1 or x = 1 having a triangle-free block graph in terms of the 2-(6, 3, 2) design with intersection numbers 1 and 2. Next we assume x > 0. We prove that a 2-(56, 16,6) design and its complement are the only QSDs having triangle-free block graph and (k − x)/(y − x) = 6.For higher values of (k − x)/(y − x), we obtain the feasible parameters of QSDs with triangle-free block graph using Mathematica.Table 4 lists the feasible parameters of QSDs with triangle-free block graph for (k − x)/(y − x) 100.We also provide other tables concerning QSDs, using Brouwer's table [6] of SRGs.
In this paper there are a large number of equations involving many variables.We used Mathematica, [41] to derive these.The reader also can use WxMaxima, [42] or any available CAS to handle the tedious calculations, which is available free on the web, if Mathematica is not available.Moreover, many background results are used and several new results are obtained.To make the paper more readable, we denote a result used the electronic journal of combinatorics 22(1) (2015), #P1.36 without proof by Result and cite the reference.The results obtained in the paper are denoted by Theorems, Propositions, etc.In addition, we have provided four Appendices at the end of this paper.In Appendix 1, we give the Mathematica code used to find feasible parameters of QSDs for given parameters (b, a, c, d) of its associated SRG, using Brouwer's tables [6].In Appendix 2, we give the Mathematica code to find feasible parameters of QSDs with triangle-free block graph for 2 m 100.In Appendix 3, we give some results about QSDs having a block graph Γ, such that Γ, the complement of Γ has no triangles.
In last Appendix 4, we obtain a result connecting SRGs, 2-distance sets in binary Hamming spaces, and QSDs.This was motivated by a recent paper of Ionin [18].
The main theorems obtained in this paper are the following: Theorem 20.For a fixed pair (c, m) or (z, m) or (λ 2, m 2), there exist only finitely many QSDs.
Theorem 21.If D is a QSD with k = 2y − x, then D is either a pair design or 2-(6, 3, 2) design or Hadamard 3-design or complement of one of these designs.

Preliminaries
In this section, we give some preliminary results needed.The reader can refer to [2] and [37] or the cited references for details.
3. The block graph of D and block graph of D, the complement of the design D, are isomorphic.
The following result giving necessary conditions on the block graph of a QSD is essentially the well known Integrality Condition for the parameters of a SRG ( [17], Theorem 7.2.4,page 220).
Lemma 5. Let D be a (v, b, r, k, λ; x, y) QSD and Γ be the (b, a, c, d) SRG block graph of D. Then, where the electronic journal of combinatorics 22(1) (2015), #P1.36 Proof.Substitute y = z + x, k = mz + x, r = nz + λ in equations ( 2) and ( 3) and observe that a−c = 2 m−m 2 −n+m n, which gives an expression for n as given in (2).Substitute this value of n in the equation ( 2) to get an expression for x as given in (1).Use Mathematica to obtain the equation given in (3).Note that m is a positive root of this quadratic.The cases 4-8 follows from above and/or equations ( 2)-( 4).

Use expression (3) to get
. Substitute this expression for b and n Now from the equation ( 1), we get (5).
Lemma 6 ([31], Theorem 2.6).For a fixed value of the block size k, there exist only finitely many QSDs with y 2.
Lemma 7 ([26], Theorem 3.2).Let D be a proper QSD with the standard parameter set (v, b, r, k, λ; x, y) with x = 0 and z = y − x = 1.Then D is a design with parameters given in ( 1) or ( 2) as follows or D is a complement of one of the design in (1).
We will need later the following classification results about SRGs.Lemma 13.Let D be a QSD whose block graph is the complete multipartite graph with s classes of size m, with the parameters (ms, m(s − 1), m(s − 2), m(s − 1)).Then Lemma 14.Let D be a QSD whose block graph is the Latin square graph LS m (n).Then . Take b = n 2 and use equation (1), to get the equation (7).
Lemma 15.Let D be a QSD whose block graph is the Steiner graph S m (n).Then, . Take b = (1+n) (m−n+m n) m and use equation (1) to get the equation (8).
Lemma 16.Let D be a QSD, with the non-zero intersection numbers x and y such that block graph of D is triangle-free.Then, where Proof.Substitute y = z + x, k = mz + x, r = nz + λ and c = 0 in equation ( 3) to obtain the following expression for b: Use (2) of Lemma 1 to get the following expression for n: Finally substitute values of k, r and b in the equation (1) to get the equation ( 9).

Finiteness results in support of Hall's conjecture
In this section, we prove a finiteness result in support of Singhi's conjecture, which is equivalent to Hall's conjecture.The main result of this section is that for a fixed pair (λ 2, m 2), there exist only finitely many QSDs.
Theorem 17.For a fixed pair (c, m), there exist only finitely many QSDs. where As before we have d − c m. Observe that the coefficient of z in ∆ is negative.Hence for a fixed pair (c, m), z is bounded by a function of (c, m).Hence x takes only finitely many values.Now use Lemma 6 to complete the proof.
Theorem 18.For a fixed pair (z, m), there exist only finitely many QSDs.
Proof.In view of Neumaier's Result 9, it is enough to show that for a fixed pair (z, m), there exist only finitely many QSDs whose block graphs are either the complete multipartite graphs with u classes of size m, with parameters (mu, m(u − 1), m(u − 2), m(u − 1)) or Latin square graphs LS m (n) or Steiner graphs S m (n).Suppose D is a QSD whose block graph is the complete multipartite graph with s classes of size m, with the parameters (ms, m(s−1), m(s−2), m(s−1)).Then the equation ( 6 As ∆ must be a perfect square, λ < . Hence for a fixed pair (z, m), λ takes finitely many values.As x satisfies the equation ( 6), x takes finitely many vales.As k = mz + x by the Lemma 6, for a fixed pair (z, m), there exist only finitely many QSDs whose block graphs are the complete multipartite graph with u classes of size m.
Suppose D is a QSD whose block graph is the Latin square graph LS m (n).Then the equation ( 7) holds.The discriminant of the quadratic (7) and Observe that (θ 1 − 1) 2 < ∆ 1 < θ 2 1 for sufficiently large n with respect to m and z.Hence n is bounded by a function of (z, m).Hence for a fixed pair (z, m), x takes only finitely many values.As before, for a fixed pair (z, m), there exist only finitely many QSDs whose block graphs are Latin square graphs.
Let D be a QSD whose block graph is the Steiner graph S m (n).Then the equation (8) holds.The discriminant of quadratic (8) for sufficiently large n with respect to m and z > 1.For z = 1, observe that either x = 0 or x = nm − m − n and for x = 0 get k = m, λ = 1.Observe that designs corresponding to x = nm − m − n are complements of designs with x = 0. Hence n is bounded by a function of (z, m).Hence for a fixed pair (m, z), x takes only finitely many values.As before for a fixed pair (z, m), there exist only finitely many QSDs whose block graphs are Steiner graphs.
Proof.For a fixed λ > 1, by (2) of Remark 4, x is bounded by λ.In view of the Theorem 18, we show that for a fixed triple (λ, x, m), z takes only finitely many values.As before, in view of the Result 9, we need to consider only three cases of QSDs, namely QSDs whose block graphs are the complete multipartite graphs with u classes of size m, with parameters (mu, m(u − 1), m(u − 2), m(u − 1)) and the Latin square graphs LS m (n) and the Steiner graphs S m (n).
If D is a QSD whose block graph is the complete multipartite graph, then parameters of D satisfy the equation (6).Hence z has finitely many choices.
Suppose D is a QSD whose block graph is the Latin square graph LS m (n).If x = 0, then from the equation ( 7) get m(n(z − 1) + 1) − 1 = 0, which is impossible.Assume x > 0, write equation ( 7) as a quadratic in n and find its discriminant ∆. 2 for sufficiently large z in terms of x and m.Hence z is bounded by some function of (x, m).
Let D be a QSD whose block graph is the Steiner graph S m (n).As before take . If x = 0, then from the equation ( 8) get m 2 (z − 1)z = 0, which implies z = 1.Observe that for these values λ = 1.Assume x > 0 and from the equation (8) Hence z is bounded by a function of (λ, m, x).

We summarize the above results as:
Theorem 20.For a fixed pair (c, m) or (z, m) or (λ 2, m 2), there exist only finitely many QSDs.

Block graphs of QSDs
In this section, we prove some results about which SRGs could be block graphs of QSDs.In the next theorem, we classify QSDs with m = 2, using the classification Result 8.In remaining cases from the fact that discriminant of the quadratic ( 5) is non-negative we get z = 1.If block graph of D is the Shrikhande graph, with parameters (16, 6, 2, 2) or the Schläfli graph, with parameters (27,16,10,8) or the Clebsch graph, with parameters (16,10,6,6), then the discriminant of the same quadratic ( 5) is not a perfect square.If the block graph of D is one of the three Chang graphs, with parameters (28, 12, 6, 4), then using the same equation ( 5) we get, x = 0 or x = 4.If x = 0, then λ = 1 and D is a 2-(8,2,1) design.The design associated with x = 4 is the complement of 2-(8, 2, 1) design.If block graph of D is the Petersen graph, with parameters (10, 3, 0, 1), then again using the same equation we get x = 1.Calculate the remaining parameters of D, to see that D is a 2-(6,3,2) design.
In the following proposition, we obtain the parameters of QSDs whose block graph is the Steiner graph S m (n) in terms of functions of (x, z, m). .Obtain v using the expression vr = bk.
The SRGs with eigenvalues a, 0 and −m (equivalently a = d) are characterized in [13] in terms of the complete multipartite graphs with s classes of size m, with parameters (ms, m(s − 1), m(s − 2), m(s − 1)) with s 2. In [13], Theorem 3.4, they proved that the only QSD with the complete s-partite block graph with parameters (2s, 2s−2, 2s−4, 2s− 2), s 2 is the Hadamard 3-design.In view of this, we give the following characterization.
In ( [13], Theorem 3.5), they ruled out the possibility of a QSD whose block graph is the Latin square graph LS 2 (n) or its complement.We rule out in Theorems 25-26, the possibility of QSD whose block graph is the Latin square graph LS m (n) or its complement, for m = 3, 4.
Theorem 25.There is no QSD whose block graph is the Latin square graph LS m (n); n m, for m = 3 and 4.
Proof.Suppose D is a QSD whose block graph is the Latin square graph LS m (n), for n m.Note that x > 0. We show that the equation ( 7) does not have integer solutions for m = 3 and 4.
Remark 27.We believe that using similar calculations as given in proofs of above theorems and using number theory, for higher values of m, the parametric classification of QSDs with block graph, the complete multipartite graph with s classes of size m or Latin square graph LS m (n) or complement of LS m (n) can be obtained.Examples of conference graphs such as Paley graphs are well known (See [7], [10], [12]).
We next rule out the possibility of a QSD whose block graph is a conference graph, using elementary arguments.
. Substitute these values of b and n in the equation ( 1) and solve for λ to get λ = . Observe that as m 2 and z 1, (2m − 1)x + (m − 1)z = 0. Now equation (1) to get P = 0, where We observe that 2m − 1 is a factor of z.Substitute z = (2m − 1)t, for positive integer t in P = 0 and get Observe that (−2m 2 + 2m − 1) (2(m−1)m(4t−1)−1) is the discriminant of this quadratic in x, which is negative, as m 2. This is a contradiction as the equation P = 0 has integer roots.

We summarize above results in the following Theorem:
Theorem 29.There is no QSD whose block graph is either a Latin square graph LS s (n) (n s) or its complement for s = 3, 4 or a conference graph.
Let Γ be a (b, a, c, d) SRG.To find feasible parameters of a QSD whose block graph is Γ, we work through the following steps: 1. m is obtained using (4) of the Lemma 5 and then n by (2).
2. As x 0, (1) of the Lemma 5 implies either Here b > a and m > 1, rules out the first possibility.
This gives us z 3. Now for each m, z and λ; x is calculated from (1) of the Lemma 5.
4. k, r and y are calculated using x, z, m, n and λ, then v and b using (1) of the Lemma 1.
For each set of parameters of a SRG other than a triangular graph or complement of a triangular graph or conference graph or Latin square graph LS m (n) for m = 2, 3, 4, or complement of LS m (n) for m = 2, 3, 4 or SRGs with smallest eigenvalue −2 listed in Table 11.12 of Brouwer [6], we execute the code given in the Appendix 1 in Mathematica to find the parameters of QSDs associated with it.As block graphs of a QSD and its complement are isomorphic we list parameters of QSDs with v 2k.We list in Table 1 parameters of QSDs whose block graphs are known.Table 2 contains the parameters of SRGs which are not block graphs of QSDs.Table 3 is the list of parameters of SRGs and associated parameters of QSDs where existence of both is unknown.There are two points to be noted, Table 2  5 Quasi-symmetric designs with triangle-free block graph We now consider QSDs, whose block graph, and not the complement of its block graph is triangle-free, as was assumed in [1].
Proposition 31.There does not exist a QSD with x = 0 and having triangle-free block graph.
In the rest of the section we assume x > 0. Proof.We substitute m = 2 and c = 0 in the equation (5).As d − c m, d = 1, 2. Using the discriminant observe that the only feasible value for z is 1.Further for z = 1 get x = 1.Now use Lemma 7 to complete the case with m = 2.
As before we substitute m = 3 and c = 0 in the equation (5).Take d = 1, 2, 3 and observe that the discriminant is either negative or not a perfect square.
Using similar calculations the possibilities of a QSD with m = 4 and having trianglefree block graph can be ruled out.For m = 5, d = 1, 2, 3, 4. Observe that the discriminant of equation ( 5) is a perfect square only for d = 2 and z = 3.Further observe that for these values of d and z, x = 3 but r is not an integer.
For m = 6, 0 < d 6. Observe that d = 4 and z = 2 are only possible values as the discriminant of equation ( 5) is a perfect square of a positive integer.For these values of d and z get x = 28 or 4. Observe that for x = 4, D is the 2-(56,16,6) design and design associated with x = 28 is the complement of 2-(56,16,6).
For higher values of m, we obtain the feasible parameters of QSDs with trianglefree block graph using Mathematica.Proof.In view of the above theorems, we take z = 1 + t and m = 3 + s for non-negative integers s, t.We observe A > 0, B 2 − 4AC − (2A(x + 2)) 2 0 and −B − 2A(x + 2) 0. Use the larger root of equation ( 9) to get λ 2(x + 2).Proof.From Proposition 35, we get λ 6. Substitute z = 1 + t, m = 3 + s for positive integers s, t in equation (9).For 2 λ 6, observe Aλ 2 + Bλ + C > 0 except for λ = 2.For λ = 2 we get (−t − 1)s 2 + (2t 2 − 2t − 6) s + 6t 2 + 5t − 5 = 0.The discriminant of this quadratic in s is t 4 + 4t  In [24], the following result is proved: D is a 3-(v, k, λ) QSD such that complement of a block graph of D is triangle-free if and only if D is a Hadamard 3-design, or D is a 3-((λ + 2)(λ 2 + 4λ + 2) + 1, λ 2 + 3λ + 2, λ), λ = 1, 2, . . .or D is a complement of one of these designs.These are two type of designs out of three possibilities obtained in the classification of quasi-symmetric 3-designs with an intersection number zero([10], Theorem 1.35).In view of this result we prove the following.Proposition 37.There does not exist a quasi-symmetric 3-design with a triangle-free block graph.A triangle-free QSD is called exceptional if x = 0 and 2y < k < y(y + 1) (equivalently 2 < m < y + 1).In [20], there is a conjecture that there are finitely many exceptional triangle-free QSDs.In support of this conjecture, it is shown in [20] that the block size k of an exceptional design D is a prime power if and only if D is a Hadamard 3-design and k = 2 n , n 2. Thus exceptional triangle-free designs do not exist for k an odd prime power.Also is shown that for a fixed value of m or y or λ with m 3 and y 2, there are finitely many such QSDs.Triangle free QSDs with x = 0 was also first investigated in [33].The classification of triangle-free QS 3-designs is given in [24].Further investigation of triangle-free QSDs with non-zero intersection numbers are carried out in [21], [25], [26], [27], [29].Considerable evidence is presented in support of the conjecture that the only triangle-free QSDs with x > 0 are the complements of QSDs with x = 0. Two results proved are summarized below: Result 41 ([21], [25], [26], [27], [29]).Let D be a triangle-free QSD with non-zero intersection numbers.Then the following hold: Remark 49.The following SRGs cannot be block graphs of QSDs: the Shrikhande graph, with parameters (16,6,2,2) or the Clebsch graph, with parameters (16,10,6,6) (see Example 5.8 and Example 5.9 in [18], Theorem 21).The paper [18] shows the importance finding bsr(Γ), which may be difficult, even if all four parameters of Γ are known.But, if all four parameters of Γ are known, then we have Mathematica program which finds feasible parameters of QSDs whose block graph is Γ.For example, can one determine bsr(S m (n)) or bsr(S 3 (n))?

Lemma 3 (
[37],Theorem 3.8).Let D be a (v, b, r, k, λ; x, y) QSD.Form the block graph Γ of D by taking as vertices the blocks of D, where two distinct vertices are adjacent whenever the corresponding blocks intersect in y points.Assume Γ is connected.Then, Γ is a SRG with parameters (b, a, c, d), where the eigenvalues of Γ are given by a = θ 0 = k(r−1)+(1−b)x y−x , θ 1 = r−λ−k+x y−x

).
Proof.As d = c + m from Lemma 5 (5), we get n = m.Solve the equation b = a + m for b to get b = m 2 z 2 +mxz+mλz+xλ x+z Now use the equation (1) to get the equation (6).

Theorem 21 .
If D is a QSD with k = 2y − x, then D is either a pair design or 2-(6, 3, 2) design or Hadamard 3-design or complement of one of these designs.Proof.Observe that, D is a QSD with k = 2y − x if and only if the smallest eigenvalue of block graph Γ of D is −2.Hence Γ is one of the graphs listed in Result 8.If block graph of D is complete n-partite graph with parameters (2n, 2n − 2, 2n − 4, 2n − 2), n 2, then by Result 10, D is the Hadamard 3-design.To rule out the possibilities of QSD whose block graph is the lattice graph, we use Result 11.If block graph of D is the triangular graph, then by Theorem 12, D is a pair design or its complement.

A quasi-symmetric 3 -
design is a QSD in which any 3-tuple of points occur in λ 3 blocks.Let D = (X, β) be a QS 3-design with intersection numbers x, y(0 < x < y) and p in X. Suppose D p = {B {p} : B ∈ β, p ∈ B} and D p = {C : C ∈ β, p ∈ C}, then D p is a 2-(v − 1, k − 1, λ 3 ) design with intersection numbers x − 1, y − 1 and D p is a 2-(v−1, k, λ−λ 3 ) design with intersection numbers x, y.Designs D p and D p are respectively called the derived and the residual designs of D. We also have λ 3 (v − 2) = λ(k − 2).

Table 3 :
is the largest and there are different QSDs having same block graph.List of feasible parameters of SRGs and associated feasible parameters of QSDs where existence of both is unknown.
Table 4 lists the feasible parameters of QSDs with triangle-free block graph for m 100 obtained by executing the code given in the Appendix 2 in Mathematica.