Walk-regular divisible design graphs

Walk-regular divisible design graphs Dean Crnkovi´c Department of Mathematics University of Rijeka, Rijeka, Croatia

and Willem H. Haemers Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands

In honor of Frank De Clerck

Abstract A divisible design graph (DDG for short) is a graph whose adjacency matrix is the incidence matrix of a divisible design. DDGs were introduced by Kharaghani, Meulenberg and the second author as a generalization of (v, k, λ)-graphs. It turns out that most (but not all) of the known examples of DDGs are walk-regular. In this paper we present an easy criterion for this to happen. In several cases walk-regularity is forced by the parameters of the DDG; then known conditions for walk-regularity lead to nonexistence results for DDGs. In addition, we construct some new DDGs, and check old and new constructions for walk-regularity. In doing so, we present and use special properties in case the classes have size two. All feasible parameter sets for DDGs on at most 27 vertices are examined. Existence is established in all but one case, and existence of a walk-regular DDG in all cases.

Keywords: divisible design graph, divisible design, walk-regular graph, (v, k, λ)-graph, Hadamard matrix. Mathematics Subject Classification: 05B05, 05E30.

1

Divisible design graphs

A graph Γ can be interpreted as a design by taking the vertices of Γ as points, and the neighborhoods of the vertices as blocks. Such a design is called a neighborhood design of Γ. The adjacency matrix of Γ is the incidence matrix of its neighborhood design. A k-regular graph on v vertices with the property that any two distinct vertices have exactly λ common neighbors is called a (v, k, λ)-graph (see [7]). The neighborhood design of a (v, k, λ)-graph is a symmetric (v, k, λ)-design. Divisible design graphs were introduced in [5] as a generalization of (v, k, λ)-graphs. Definition 1.1 A k-regular graph on v vertices is a divisible design graph (DDG for short) with parameters (v, k, λ1 , λ2 , m, n) whenever the vertex set can be partitioned into m classes of size n, such that two distinct vertices from the same class have exactly λ1 common neighbors, and two vertices from different classes have exactly λ2 common neighbors.

1

Note that a DDG with m = 1, n = 1, or λ1 = λ2 is a (v, k, λ)-graph. In this case we call the DDG improper, otherwise it is proper. An incidence structure with v points and constant block size k is a (group) divisible design with parameters (v, k, λ1 , λ2 , m, n) whenever the point set can be partitioned into m classes of size n, such that two vertices from the same class have exactly λ1 common neighbors, and two vertices from different classes have exactly λ2 common neighbors. It is clear that the neighborhood design of a DDG is a divisible design. Conversely, a divisible design admitting a symmetric incidence matrix with zero diagonal is the neighborhood design of a DDG.

2

The adjacency and the quotient matrix

Here we briefly survey some relevant properties of DDGs. For proofs and more details we refer to [5]. As usual, we denote by I` (or just I) and J` the ` × ` identity and all-ones matrix, respectively. The all-ones matrix and vector of undetermined size are denoted by J and 1. Let us define K(m,n) = Im ⊗ Jn . Then A is the adjacency matrix of a DDG with parameters (v, k, λ1 , λ2 , m, n) if and only if there is a vertex ordering such that AA> = A2 = kIv + λ1 (K(m,n) − Iv ) + λ2 (Jv − K(m,n) ).

(1)

From this it readily follows that A has the following eigenvalues p p k, θ1 = k − λ1 , θ2 = −θ1 , θ3 = k 2 − λ2 v, θ4 = −θ3 with multiplicities 1, f1 , f2 , g1 , g2 , respectively, where f1 + f2 = m(n − 1), g1 + g2 = m − 1, and k + (f1 − f2 )θ1 + (g1 − g2 )θ3 = trace A = 0. Note that the parameters of a DDG do not always determine the multiplicities. But they do as soon as one more equation holds. This is the case if θ1 or θ3 is not an integral, as follows from the following result ([5], Theorem 2.2). Theorem 2.1 For a proper DDG the following holds. (a) k − λ1 or k 2 − λ2 v is a nonzero square. (b) If k − λ1 is not a square, then f1 = f2 = m(n − 1)/2. (c) If k 2 − λ2 v is not a square, then g1 = g2 = (m − 1)/2. In general, a proper DDG will have five distinct eigenvalues, although in specific cases some eigenvalues may coincide or have zero multiplicity. A connected DDG with at most three distinct eigenvalues is a (v, k, λ)-graph, therefore a connected proper DDG has four or five distinct eigenvalues. The vertex partition from the definition of a DDG gives a partition (called the canonical partition) of the adjacency matrix A = (Aij )1≤i,j≤m . Bose [1] proved that the canonical partition of a symmetric divisible design is an equitable partition (also called tactical decomposition) of its incidence matrix A, which means that each block Aij has constant row and column sum (see [5] for a short proof). This allows for the definition of the matrix R = [rij ], where rij is the row (and column) sum of Aij . The matrix R is called the quotient matrix of A. 2

Theorem 2.2 Consider a proper DDG with parameter set (v, k, λ1 , λ2 , m, n) and eigenvalues k, ±θ1 , ±θ3 with respective multiplicities 1, f1 , f2 , g1 , g2 . Then the canonical partition is equitable. The quotient matrix R = [rij ] is symmetric and satisfies R1 = k1 , R2 = (k 2 − λ2 v)Im + λ2 nJm . √ The eigenvalues of R are k, θ3 = k 2 − λ2 v and θ4 = −θ3 with multiplicities 1, g1 and g2 , respectively. Moreover, rii ≤ n − 1, and rii is even if n is odd for i = 1, . . . , m. Note that the eigenvalues of R follow from the given equations, and the conditions on rii follow because Aii is the adjacency matrix of a graph with n vertices, which happens to be regular of degree rii .

3

Walk-regular graphs

A graph is walk-regular if for every ` ≥ 2 the number of closed walks of length ` at a vertex x is independent of the choice of x (see [4]). In terms of the adjacency matrix A, this can be rephrased as A` has constant diagonal for every positive integer `. Note that walkregularity implies regularity (take ` = 2). Examples of walk-regular graphs are strongly regular graphs, and vertex-transitive graphs. It turns out that many (but not all) known DDGs are walk-regular. In this section we investigate this phenomenon. First we quote a well-known characterization and its corollary (see for example [2], Chapter 15). Lemma 3.1 Let Γ be a connected graph whose adjacency matrix A has r distinct eigenvalues. Then Γ is walk-regular if and only if A` has constant diagonal for 2 ≤ ` ≤ r − 2. If Γ is regular, A2 has constant diagonal, hence: Corollary 3.2 A connected regular graph with at most four distinct eigenvalues is walkregular. A DDG has at most five distinct eigenvalues, but in many cases it has only four, which makes the DDG walk-regular, provided it is connected. In a disconnected DDG λ2 = 0 and each component is either an (n, k, λ)-graph, or the incidence graph of a symmetric (n, k, λ)-design (see [5], Proposition 4.7). It easily follows that such a disconnected DDG is not walk-regular if the mentioned types of components are both present, and walk-regular otherwise. A DDG with five distinct eigenvalues can also be walk-regular. To decide on this the following characterization is useful. Theorem 3.3 A proper DDG is walk-regular if and only if the quotient matrix R has constant diagonal. Proof. By the above remark, we only need to consider connected DDGs. By Lemma 3.1, if suffices to show that R has constant diagonal if and only if A3 has constant diagonal. Formula (1) implies A3 = (k − λ1 )A + (λ1 − λ2 )AK(m,n) + λ2 kJv . From the fact that the canonical partition is a tactical decomposition with quotient matrix R, it follows that AK(m,n) = K(m,n) A = R ⊗ Jn . Therefore A3 − (λ1 − λ2 )(R ⊗ Jn ) has constant diagonal, which proves our claim.  √ 2 In particular, if a DDG is walk-regular, then trace R = k + (g1 − g2 ) k − λ2 v is divisible by m and divisible by 2m if n is odd. 3

Theorem 3.4 A DDG for which θ1 =

√

k − λ1 is not an integer, is walk-regular.

Proof. If k − λ1 is not a square, then Theorem 2.1 gives f1 = f2 and hence 0 = trace A = k + (f1 − f2 )θ1 + (g1 − g2 )θ3 = k + (g1 − g2 )θ3 = trace R. Therefore R has zero diagonal.



By counting closed walks in a graph Γ one obtains necessary conditions for walk-regularity in terms of the eigenvalues of Γ (see for example [3]). Proposition 3.5 Suppose Γ is walk P regular with eigenvalues λ1 ≥ . . . ≥ λv . Then for v every positive integer ` the number v1 i=1 λ`i is an integer, which is even if ` is odd. 1 ` Proof. P ` The number of closed walks of length ` at any vertex x of Γ equals v traceA = 1 λi . If ` is odd, a closed walk of length ` at a vertex x is different from the walk in v opposite direction. Therefore the total number of such walks is even. 

With Corollary 3.2 and Theorem 3.4, the conditions of Proposition 3.5 lead to nonexistence results for DDGs. For example: Theorem 3.6 If for a DDG k − λ1 is not a square, then kλ2 is even. Proof. ByPTheorem 3.4 the graph √ is walk-regular and Theorem 2.1√implies f1 = f2 . From 0 = λi = k + (g1 − g2 ) k 2 − λ2 v we obtain g1 − g2 = −k/ k 2 − λ2 v. Hence P 3 3 λi = k 3 + (g1 − g2 )(k 2 − λ2 v) 2 = vkλ2 . So kλ2 is even by Proposition 3.5.  Corollary 3.7 There exists no DDG with parameters (24, 9, 4, 3, 6, 4), (24, 9, 6, 3, 12, 2) and (24, 15, 12, 9, 12, 2). Theorem 3.8 Suppose Γ is a DDG for which k 2 − λ2 v is not a square. If m = 3, or m does not divide k, then Γ is not walk-regular. Proof. Assume Γ is walk-regular. Since k 2 − λ2 v is not a square, trace R = k. Hence m divides k. If m = 3, there is only one possible quotient matrix with row sum k and constant diagonal: R = 31 kJ3 . But then R has an eigenvalue 0, so k 2 − λ2 v = 0, which is a contradiction.  For example a DDG with parameters (24u2 , 8u2 + 2u, 4u2 + 2u, 2u2 + u, 3, 8u2 ) is not walkregular, because m = 3 and k 2 − λ2 v = 4u2 (4u2 + 2u + 1) which is not a square. The DDGs of Construction 4.9 from [5] have these parameters, and so are examples of connected proper DDGs that are not walk-regular. All other connected DDGs constructed in [5] are walkregular. This holds in particular for Construction 4.8 from [5], which is a special case of the following construction. Theorem 3.9 Let H be a regular graphical Hadamard matrix of order 4u2 with diagonal entries −1 and row sum 2u (u can be negative), and let D be the adjacency matrix of an (n, k 0 , λ)-graph. Replace each entry −1 of H by D, and each +1 by Jn − D. Then we obtain the adjacency matrix A of a walk-regular DDG with parameters (4nu2 , 2nu2 + u(n − 2k 0 ), 4λu2 + u(2u + 1)(n − 2k 0 ), nu2 + u(n − 2k 0 ), 4u2 , n). Proof. By straightforward verification if follows that A satisfies Equation (1). Moreover, all diagonal blocks of A have equal row sum (= k 0 ). Therefore the obtained DDG is walkregular by Theorem 3.3.  4

If we take D = Jn −I we obtain Construction 4.8 from [5]. In case u = 1, and k 0 = 1 or k 0 = n−1, the above parameters become (4n, n+2, n−2, 2, 4, n) and (4n, 3n−2, 3n−6, 2n−2, 4, n), respectively. In both cases k − λ1 = 4 is a square, so it may be possible that there exist a DDG with these parameters which is not walk-regular. It turns out that this is indeed the case when n is even. Theorem 3.10 If n is even and n ≥ 6, then there exists a connected proper DDG with parameters (4n, n + 2, n − 2, 2, 4, n) and one with parameters (4n, 3n − 2, 3n − 6, 2n − 2, 4, n), neither of which is walk-regular. Proof. Let Z denote the reverse identity matrix of order n (that is (Z)i,j = 1 if i+j = n+1, and (Z)i,j = 0 otherwise). Then Z is symmetric with zero diagonal and Z 2 = In . Define Z = Jn − Z, I = In and I = Jn − In . Then     I I I I I I I I  I I I I   I I I I       I I Z Z  and  I I Z Z  I I Z Z I I Z Z are adjacency matrices of DDGs with the required parameters. Obviously the diagonal entries of each quotient matrix take two values: 1 and n − 1. So the graphs are not walkregular. Moreover, n ≥ 6 implies λ1 > λ2 > 0, therefore the DDGs are connected and proper. 

4

Thin divisible design graphs

A DDG with n = 2 is called thin. Thin DDGs have special properties. It is clear that there are only four possible blocks in the canonical partition, being:         0 0 0 1 1 0 1 1 , , and . 0 0 1 0 0 1 1 1 From this we see that such a graph has a nontrivial automorphism, because interchanging the two vertices in each class is an automorphism. Proposition 4.1 A thin DDG has an involution, and the orbits are the m classes. Another observation is that the diagonal entries of the quotient matrix R are 0 or 1. Therefore the number of entries equal to one equals trace R, so Theorem 3.3 gives: √ Proposition 4.2 A thin DDG is walk-regular if and only if k + (g1 − g2 ) k 2 − λ2 v equals 0 or m. In the case of thin DDGs, the quotient matrix R has a natural partner Q of the same size. If Aij = [ ab ba ] is a block of A, then the corresponding element of Q is defined by (Q)i,j = a − b. We have that (R)i,j = a + b, so Q ≡ R (mod 2). The entries of R are 0, 1 or 2, and those of Q are −1, 0, or 1. The following result extends Theorem 2.2 for thin DDGs. Theorem 4.3 Consider a proper thin DDG with quotient matrix R and partner Q. √ (a) The eigenvalues of R are k, θ3 = k 2 − λ2 v, and θ4 = −θ3 with multiplicities 1, g1 and g2 = m − g1 − 1, respectively.

5

(b) The eigenvalues of Q are θ1 = f2 = m − f1 , respectively.

√

k − λ1 , and θ2 = −θ1 with multiplicities f1 and

Proof. Case (a) follows from Theorem 2.2. Suppose u is an eigenvector of R with eigenvalue θ, then u ⊗ [ 1 1 ]> is an eigenvector of the adjacency matrix A with the same eigenvalue θ. Therefore the eigenvalues of A equal to k, θ3 and θ4 have eigenvectors which are constant over the blocks of the canonical partition. Next assume v is an eigenvector of Q with eigenvalue θ0 . Then w = v ⊗ [ 1 −1 ]> is an eigenvector of A with the same eigenvalue θ0 . Clearly the eigenvector w of A is orthogonal to the eigenvectors of A for the eigenvalues k and ±θ3 . Therefore θ0 = ±θ1 , which implies (b).  The spectrum of the adjacency matrix A is the union of the spectrum of R and the spectrum of Q. Theorem 4.3 implies that Q2 = (k − λ1 )I. A square (−1, 0, 1)-matrix with these properties is called a weighing matrix of weight k − λ1 . So Q is a weighing matrix, which is symmetric and has all its diagonal entries equal to −1 or 0. Theorem 4.4 Let Q be a symmetric weighing matrix of order m and weight w satisfying (Q)i,i 6= 1 (i = 1, . . . , m). Let R be a symmetric (0, 1, 2)-matrix, satisfying R ≡ Q (mod 2), (R)i,i 6= 2 (i = 1, . . . , m), and R2 = αI + βJ for some α, β ∈ R. Then   1 Q+R R−Q A= 2 R−Q Q+R is the adjacency matrix of a thin DDG with quotient matrix R, partner Q, and parameters p (v = 2m, k = α + mβ, λ1 = k − w, λ2 = β/2, m, 2). Proof. It follows by straightforward verification that A satisfies Equation (1). Here we of course note the small caveat that vertices are not canonically ordered: The two vertices of class i are in row (and column) i and m + i.  The above theorem helps in the construction and non-existence results of thin DDGs, as we shall see below. Theorem 4.5 If there exist a regular graphical Hadamard matrix of order 4u2 with row sum 2u, and a Hadamard matrix of order 2u2 , then there exists a walk-regular DDG with parameters (24u2 , 12u2 − 2u, 4u2 − 2u, 6u2 − 2u, 12u2 , 2). Proof. Let H0 be a regular graphical Hadamard matrix of order 4u2 with row sum 2u, and let H1 and H2 be Hadamard matrices of order 4u2 and 2u2 , respectively. (H1 does not have to be related to H0 .) Define H0 = H0 + J, H1 = H1 ⊗ [ 1 −1 ] , and H2 = H2 ⊗ J2 . Then     O H1 H0 J J   R =  J H0 J  , and Q =  > O H2  H1 > J J H0 H2 O satisfy the conditions of Theorem 4.4. Clearly (R)i,i = 0 for i = 1, . . . , 12u2 , which implies walk-regularty.  There exists a Hadamard matrix of order 2, a graphical Hadamard matrix of order 4 with row sum 2, and also one with row sum −2. Hence: Corollary 4.6 There exists walk-regular DDGs with parameters (24, 10, 2, 4, 12, 2) and (24, 14, 6, 8, 12, 2).

6

Next we make use of R and Q for a nonexistence result. Proposition 4.7 There exists no DDG with parameters (26, 9, 0, 3, 13, 2). Proof. Since λ1 = 0, the quotient matrix R is a (0, 1) matrix of order 13, satisfying RR> = 6J + 3I. Therefore R is the incidence matrix of a 2-(13, 9, 6) design. This means that the complement N = J − R is a symmetric incidence matrix of the projective plane of order 3. It is well know that (up to simultaneous reordering of rows and columns) N is unique. Define S = [ 1 I3 ], then   2I4 O S > J −S >  O  J3 I3 I3 . N +I =  S  I3 I3 I3 J −S I3 I3 I3 Since 2I4 ≡ O (mod 2), it follows that 2-rank(N + I) ≥ 2-rank S + 2-rank J3 + 2-rank S > = 7. Because N = J − R, we have 2-rank(R + I) ≥ 6. Next we consider Q. By Proposition 4.3, Q has eigenvalues ±3 with multiplicities f1 and f2 = 13 − f1 , respectively. But k 2 − λ2 v = 3 is not a square, so g1 = g2 = 6, and trace A = 0 gives f1 = 5, and f2 = 8. Hence rank(Q+3I) = 5, but R +I ≡ Q+3I (mod 2), so 6 ≤ 2-rank(R + I) = 2-rank(Q + 3I) ≤ rank(Q + 3I) = 5, contradiction.  We remark that a divisible design with parameters (26, 9, 0, 3, 13, 2) and a symmetric incidence matrix does exist (points and blocks are the nonzero vectors in F33 ; incident if the inner product equals 1). It is the zero diagonal that causes a contradiction. Not all thin DDGs are walk-regular, as is shown by the next example. Proposition 4.8 There exists a DDG with parameter set (20, 9, 0, 4, 10, 2), which is not walk-regular. Proof. Take         Q=       

−1 0 1 1 1 1 1 −1 −1 1

0 1 1 1 1 −1 1 1 1 1 1 −1 0 1 1 1 0 −1 1 1 1 1 1 −1 0 1 1 1 0 −1 −1 −1 1 −1 1 1 1 −1 −1 1 1 −1 1 1 −1 −1 1 −1 1 −1

1 −1 −1 1 −1 1 0 −1 −1 −1

−1 1 1 −1 −1 1 −1 0 −1 −1

−1 1 −1 1 1 −1 −1 −1 0 −1

1 −1 1 −1 1 −1 −1 −1 −1 0

        ,       

and take R = |Q|. Then Q and R satisfy Theorem 4.4. Clearly the quotient matrix does not have constant diagonal, so the graph is not walk-regular.  This construction appeared already in Master’s thesis of M.A. Meulenberg [6]. It turns out (see next section) that 20 is the smallest number of vertices for which there exists a connected DDG that is not walk-regular1 . A walk-regular example with the above parameters also exists and is given by the distance-regular Johnson graph J(6, 3), see [5]. 1 This DDG was presented for the first time at the conference IPM 20–Combinatorics 2009 in Tehran as a present for the 20th anniversary of the IPM.

7

5

Small parameters

For many parameter sets the above conditions imply that a DDG with those parameters is walk-regular. These include most parameter sets with at most 27 vertices which are presented in Table 1. For all parameter sets on at most 27 vertices we are able to decide if there exist a walk-regular DDG or not, and for all but one parameter set we are able to decide if there exits a DDG which is not walk-regular. Most cases follow from one of the results above. A few cases need special attention. The first proposition appeared in [6]. Proposition 5.1 Any DDG with one of the following parameters sets is walk-regular. (12, 5, 1, 2, 4, 3), (12, 7, 3, 4, 4, 3), (15, 4, 0, 1, 5, 3), (20, 7, 3, 2, 4, 5), (20, 13, 9, 8, 4, 5). Proof. Assume there exist a DDG with parameters (12, 5, 1, 2, 4, 3) which is not walkregular. Then the eigenvalues are 5, 2, −2, 1 and −1. The respective multiplicities f1 , f2 , g1 and g2 are all positive (otherwise the graph would be walk-regular). This gives only one possibility (f1 , f2 , g1 , g2 ) = (3, 5, 1, 2). Hence trace R = 4. By Theorem 3.3 of [5], the entries of the quotient matrix R can take only two values: 1 and 2. Thus all diagonal entries are equal to 1, contradiction. Assume there exist a DDG with parameters (12, 7, 3, 4, 4, 3) which is not walk-regular. Again there is only one possible spectrum with five distinct eigenvalues, being: 7, 2, −2, 1, −1 with multiplicities 1, 2, 6, 2 and 1. Hence trace R = 8. The diagonal entries of R are at most n − 1 = 2, so they must all be equal to 2, contradiction. Assume there exist a DDG with parameters (15, 4, 0, 1, 5, 3) which is not walk-regular. The only possibility for five distinct eigenvalues is 4, 2, −2, 1 and −1 with multiplicities 1, 4, 6, 2 and 2, respectively. Hence trace R = 4. Since λ1 = 0, the entries of the quotient matrix R can take only two values: 0 and 1. Thus four diagonal entries of R are equal to one, which is impossible because n is odd. Assume there exist a DDG with parameters (20, 7, 3, 2, 4, 5) which is not walk-regular. The only possible spectrum with five distinct eigenvalues is 7, 2, −2, 3, −3 with multiplicities 1, 7, 9, 1, 2. Hence trace R = 4. Since n is odd, the diagonal entries of R are even. Using R1 = 7 1 and R2 = 10J + 9I we find only one feasible quotient matrix   4 1 1 1  1 0 3 3   R=  1 3 0 3 . 1 3 3 0 However it turns out to be impossible to built a DDG with this R (see [6] for details). Assume there exist a DDG with parameters (20, 13, 9, 8, 4, 5) which is not walk-regular. The only possible spectrum with five distinct eigenvalues is 13, 2, −2, 3, −3 with multiplicities 1, 4, 12, 2, 1. Hence trace R = 16. The diagonal entries of R are at most 4, so they are all equal to 4, contradiction.  We also found one more sporadic DDG: Proposition 5.2 There exists a walk-regular DDG with parameters (24, 10, 3, 4, 8, 3).

8

Proof. The following matrix                  A=                

0 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0

0 0 0 1 0 1 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1

0 0 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0

1 1 0 0 0 0 1 1 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1

1 0 1 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 1 0 0 1 0

0 1 1 0 0 0 0 1 1 0 1 1 0 0 1 0 0 1 1 0 0 1 0 0

1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 0 1 0

1 0 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 1 0 0

0 1 1 0 1 1 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0 0 1

1 1 0 1 1 0 0 1 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1

1 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0

0 1 1 0 1 1 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0

1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 1 0 1

0 0 1 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 1 0 1 0 1 1

0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 1 1 1 1 0

1 0 0 0 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 1 0

0 0 1 1 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 1

0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 1 0 0 1 1

1 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 0 1 0 0 0 1 0 1

0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 0 0 1 1

0 1 0 1 0 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0

1 0 0 0 0 1 0 1 0 0 0 1 1 0 1 1 1 0 1 0 1 0 0 0

0 0 1 0 1 0 1 0 0 0 1 0 0 1 1 1 0 1 0 1 1 0 0 0

0 1 0 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 0 0 0

                                 

satisfies Equation (1). Moreover all diagonal blocks of A are O. Therefore A is the adjacency matrix of a walk-regular DDG.  Proposition 5.3 There exists no DDG with parameters (25, 12, 8, 5, 5, 5). Proof. The quotient matrix R of such a DDG satisfies R1 = 12 1 and RR> = 19I5 + 25J5 . Therefore, each row sum of R equals 12, the inner product of any row of R with itself equals 44. It follows that the entries of a row of R can only take values from the multiset {5, 4, 1, 1, 1} or {5, 3, 3, 1, 0}. Since R is symmetric, R has a diagonal entry equal to 5, which is impossible by Theorem 2.2.  Proposition 5.4 There exists no DDG with parameters (27, 6, 3, 1, 9, 3). Proof. The quotient matrix R satisfies R1 = 6 1 and R2 = 9I9 + 3J9 . Therefore, each row sum of R equals 6, the inner product of any row of R with itself equals 12, and any two distinct rows of R have inner product 9. It follows that the entries of a row of R can take values from the multiset {2, 2, 2, 0, 0, 0, 0, 0, 0} and {3, 1, 1, 1, 0, 0, 0, 0, 0}. A row with the entries from the first multiset can not have inner product 9 with any other row, therefore all entries of R must have the entries from the second multiset. Since R is symmetric, at least one diagonal entry equals 3, which contradicts Theorem 2.2.  In [5] the authors generated all putative parameters sets (v, k, λ1 , λ2 , m, n) for proper DDGs on at most 27 vertices. Except for the sets corresponding to the straightforward cases λ1 = k, λ2 = 0 and λ2 = 2k − v, they obtained fifty parameter sets and for each set they tried to decide on existence or nonexistence of DDGs. In ten cases existence remained undecided. In this paper we have resolved nine of these ten cases. Moreover, we decided for which parameter sets there exits a walk-regular DDG and a non-walk-regular DDG. We copied this information in Table 1, included the new results, but deleted the parameter sets for which no DDG exits (proved in [5], Corollary 3.7, Proposition 4.7, 5.3 or 5.4). In the table multiplicities of the eigenvalues (denoted as exponents) are only given if they are determined by the parameters. The column ‘WR’ indicates if there exists a walk-regular DDG, and the column ‘notWR’ indicates existence of a DDG which is not walk-regular. 9

v

k

λ1

λ2

m

n

θ1 1

f

θ2 2

8 10 12 12 12 12 15 18 18 20 20 20 20 20 24 24 24 24 24 24 24 24 24 26 27 27

4 5 5 5 6 7 4 9 9 7 7 9 13 13 6 7 8 10 10 10 14 14 16 13 16 18

0 4 0 1 2 3 0 6 8 3 6 0 9 12 2 0 4 2 3 6 6 7 12 12 12 9

2 2 2 2 3 4 1 4 4 2 2 4 8 8 1 2 2 4 4 3 8 8 10 6 9 12

4 5 6 4 3 4 5 6 9 4 10 10 4 10 3 8 4 12 8 3 12 8 4 13 9 9

2 2 2 3 4 3 3 3 2 5 2 2 5 2 8 3 6 2 3 8 2 3 6 2 3 3

21 √ 3 5 2 23 2 2 √ 6 3 2 3 2 29 √ 8 7 2 √ 6 8 √ 8 7 28 √ 6 8 √ 8 7 2 25 36

−23 −15 √ 3 − 5 −2 −26 −2 −2 √ 6 − 3 −19 −2 −110 −3 −2 −110 −212 √ 8 − 7 −2 √ 6 − 8 √ 8 − 7 −213 √ 6 − 8 √ 8 − 7 −2 −113 −213 −312

f

g

θ3 1 03 √ 2 5 1 02 1 1 31 34 3 35 1 3 34 √ 1 12 4 23 21 √ 1 28 22 4 √ 6 13 √ 4 13 08

g

θ4 2 √ 2 − 5 −15 −1 −1 −1 −34 −34 −3 −34 −1 −3 −35 √ 1 − 12 −17 −4 −28 −26 √ 1 − 28 −29 −27 −4 √ 6 − 13 √ 4 − 13 -

WR

notWR

reference

yes yes yes yes yes yes yes yes yes yes yes yes yes yes no yes yes yes yes no yes yes yes yes no yes

no no no no no no no no no no no yes no no yes no yes no no yes no no yes no

[5], 3.2 [5], 3.2 [5], 3.2 [5], 5.1 [5], 3.2 [5], 5.1 [5], 5.1 [5], 3.4 [5], 3.2 [5], 5.1 [5], 3.2 [5], 4.8 [5], 5.1 [5], 3.2 3.8, [5] [5], 3.2 [5], 3.10 4.6, 3.4 5.2, 3.4 3.8, [5] 4.6, 3.4 [5], 3.2 [5], 3.10 [5], 3.2 3.8, – [5], 3.2

no

Table 1: Feasible parameters for proper DDGs with v ≤ 27, 0 < λ2 < 2k − v, λ1 < k.

References [1] R.C. Bose, Symmetric group divisible designs with the dual property, J. Stat. Plann. Inference 1 (1977), 87–101. [2] A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer 2012. [3] E.R. van Dam, Graphs with four eigenvalues, Linear Algebra Appl. 226-228 (1995), 139–162. [4] C.D. Godsil and B.D. McKay, Feasibility conditions for the existence of walk-regular graphs, Linear Algebra Appl. 30 (1980), 51–61. [5] W.H. Haemers, H. Kharaghani, M.A. Meulenberg, Divisible Design Graphs, J. Combin. Theory Ser. A 118 (2011), 978–992. [6] M.A. Meulenberg, Divisible Design Graphs, Master’s thesis, Tilburg University, 2008. [7] A. Rudvalis, (v, k, λ)-graphs and polarities of (v, k, λ)-designs, Math. Z. 120 (1971), 224–230.

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