Abstract
Designs over edgeregular, coedgeregular and amply regular graphs are investigated. Using linear algebra, we obtain lower bounds in certain inequalities involving the parameters of the designs. Some results on designs meeting the bounds are obtained. These designs are over connected regular graphs with least eigenvalue \(2\), have the minimal number of blocks and do not appear in an earlier work. Partial classification such designs over strongly regular graphs with least eigenvalue \(2\) is given.
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1 Introduction
Let \(\Gamma =(V,E)\) be a regular graph of valency d, having vertex set V and edge set E. Let \(v=V\), and let \(k,\lambda \) and \(\mu \) be nonnegative integers with \(2\le k \le v1\) and \(\lambda \ne \mu \). A \((v,k,\lambda ,\mu )\)design over \(\Gamma \) is a pair \(D=(V,{\mathcal {B}}),\) where \({\mathcal {B}}\) is a set of ksubsets of V, called blocks, such that if \(i,j\in V, i\ne j\), then there are exactly \(\lambda \) blocks containing \(\{i,j\}\) if \(\{i,j\}\in E\) and exactly \(\mu \) blocks, containing \(\{i,j\}\) if \(\{i,j\}\not \in E\). The notion of a \((v,k,\lambda ,\mu )\)design over a regular graph \(\Gamma \) occurs in [6]. In case \(\Gamma =K_v\), the complete graph on v vertices, then a \((v,k,\lambda ,\mu )\)design is same as a 2\((v,k,\lambda )\) design. In case, \(\Gamma \) is a strongly regular graph (SRG), and then, \((v,k,\lambda , \mu )\)designs are precisely the 2class partially balanced designs (PBIBDs) with association scheme \(\Gamma \) of Bose and Nair [1].
For any \((v,k,\lambda ,\mu )\)design \(D=(V,{\mathcal {B}})\) over a regular graph \(\Gamma =(V,E)\), we list together some results proved in [5] and [6] which will be used in the sequel.
Result 1.1

1.
Each point of V lies in a constant number r (called replication number) of blocks and that \(bk=vr\), where \(b={\mathcal {B}}\).

2.
The inequality \(r\ge 2\lambda \mu \) holds, and designs with \(r=2\lambda \mu \) have the following combinatorial property:

P1: For any three vertices \(x,y,z \in V\) such that \(\{x,y\},\{x,z\} \in E\) and \(\{y,z\} \notin E\),

(a)
any block containing x must contain y or z and

(a)
any block containing y and z must contain x.

(a)


3.
Let A be the usual \(v\times b\) (0, 1) adjacency matrix of \(\Gamma .\) If \(\dfrac{r\mu }{\mu \lambda }\) is not a multiple eigenvalue of A, then \(b\ge v.\)

4.
If \(r=2 \lambda \mu \), then \(\dfrac{r\mu }{\mu \lambda }=2\). Furthermore, let \(\Gamma \) be a connected regular graph with \(\dfrac{r\mu }{\mu \lambda }\) an eigenvalue of \(\Gamma \) other than the valency. Then, the \((v,k,\lambda ,\mu )\) design over \(\Gamma \) has another combinatorial property.

P2: For any pointblock pair (x, B), the number of neighbors of x in B depends only on whether \(x\in B\) or not.

Therefore, if \(\Gamma \) is a connected strongly regular graph with smallest eigenvalue \(2\) and D is a \((v,k,\lambda ,\mu )\)design over \(\Gamma \) with \(r=2\lambda \mu \), then we can use both the above combinatorial properties P1 and P2.
In a fundamental result, Seidel [8] found all the connected strongly regular graphs with the smallest eigenvalue \(2\).
Theorem 1.2
(Seidel [8]) Let \(\Gamma \) be a strongly regular graph with smallest eigenvalue \(2\). Then, \(\Gamma \) is one of the followings

1.
the complete npartite graph \(K_{n\times 2}\), with parameters \((2n,2n2,2n4,2n2), n \ge 2\),

2.
the lattice graph \(L_2(n)\), with parameters \((n^2,2(n 1),n2,2), n \ge 3\),

3.
the Shrikhande graph, with parameters (16, 6, 2, 2),

4.
the triangular graph T(n), with parameters \( (n(n1)/2,2(n2),n2, 4), n \ge 5 \),

5.
one of the three Chang graphs, with parameters (28, 12, 6, 4),

6.
the Petersen graph, with parameters (10, 3, 0, 1),

7.
the Clebsch graph, with parameters (16, 10, 6, 6),

8.
the Schl\(\ddot{a}\)fli graph, with parameters (27, 16, 10, 8).
In [5], Seidel’s theorem was used to obtain a complete classification of \((v,k,\lambda ,\mu )\)designs over connected strongly regular graphs with smallest eigenvalue \(2\) and having the least number of blocks. Table 1 displays all the possible designs arising.
At the conclusion of [5], the authors remarked that the methods developed therein may possibly extend to other regular graphs such as edgeregular, amply regular, coedgeregular and trianglefree graphs. This is motivation of the present paper in which we investigate minimal designs over the above types of graphs.
In the present paper, we restrict the notion of designs over regular graphs and consider designs over edgeregular, coedgeregular and amply regular graphs. We then apply the methods developed in the paper [5] to obtain certain inequalities involving the parameters of the designs. Some results on designs meeting the lower bounds of the inequalities are obtained. These designs have the least number of blocks over connected regular graphs with smallest eigenvalue \(2\) and do not appear in the classification of [5]. Our aim is to classify such designs satisfying the lower bounds of inequalities having least eigenvalue \(2\). Partial classification of such designs over strongly regular graphs with least eigenvalue \(2\) is given. To find cliques or cocliques of a graph and also to perform other graph operations, we used computer algebra system SageMath [10]. For graph G, L(G) denotes the line graph of G and \({\overline{G}}\) denotes the complement of G.
2 Preliminaries
We refer to [2] for graph theoretic concepts in detail. In what follows, we shall first consider regular graphs having one or more of the following properties:
 R1::

Any two adjacent vertices have exactly \(\lambda =\lambda (\Gamma )\) common neighbors.
 R2::

Any two vertices at a distance 2 have exactly \(\mu =\mu (\Gamma )\) common neighbors.
 R3::

Any two nonadjacent vertices have exactly \(\mu =\mu (\Gamma )\) common neighbors.
It is clear that R2 follows from R3, and these two properties are equivalent if \(\Gamma \) is connected with diameter at most 2.
A regular graph on v vertices and valency d is called edgeregular with parameters \((v,d,\lambda )\) if R1 holds, amply regular with parameters \((v,d,\lambda ,\mu )\) if R1 and R2 hold, coedgeregular with parameters \((v,d,\mu )\) if R3 holds, and strongly regular with parameters \((v,d,\lambda ,\mu )\) if R1 and R3 hold.
If \(\gamma \) is any vertex of \(\Gamma \), then we write \(\Gamma (\gamma )\) for the set of neighbors of \(\gamma \) in \(\Gamma \), and, more generally \(\Gamma _i(\gamma )\) for the set of vertices at distance i from \(\gamma \) in \(\Gamma \). Also, \(\gamma \sim \delta \) denotes for vertices \(\gamma \) and \(\delta \) of \(\Gamma \) that \(\gamma \) is adjacent to \(\delta \), while \(\gamma ^{\perp }\) is the set \(\{\gamma \}\cup \Gamma (\gamma )\), and for \(i,j \in V\), \(i\ne j\), d(i, j) denotes the distance between i and j.
Definition 2.1
Let \(\Gamma =(V,E)\) be finite, undirected graph with neither loops nor multiple edges. A coclique (independent) set in a graph \(\Gamma \) is subgraph in which no two vertices are adjacent. A maximum coclique is a coclique of largest cardinality. The independence number \(\alpha (\Gamma )\) of \(\Gamma \) is the cardinality of a largest coclique (independent set) in \(\Gamma \). A clique in \(\Gamma \) is a complete subgraph of \(\Gamma \). A maximum clique is a clique of largest cardinality.
Proposition 2.2
([2], Proposition 1.3.2.; Hoffman [4])
Let \(\Gamma \) be a regular graph with v vertices, valency d and with smallest eigenvalue \(m\).

1.
If C is a coclique of \(\Gamma \), then \(C \le (1+d/m)^{1}v\), with equality if and only if every vertex \(\gamma \notin C\) has the same number of neighbors (namely m)in C.

2.
If \(\Gamma \) is strongly regular and C is a clique of \(\Gamma \), then \(C\le (1+d/m)\), with equality if and only if every vertex \(\gamma \notin C\) has the same number of neighbors (namely \(\mu /m\)) in C.
Example 2.3

1.
The complement \(\Gamma \) of an \(n \times (n + 1)\)grid \(\Delta \) is an edgeregular graph with \(k(\Gamma ) = n(n 1), \lambda (\Gamma ) = (n 1)(n  2)\); thus, we have an infinite family of edgeregular graphs with \(\lambda = k +1 \sqrt{4 k + 1}\) which are not strongly regular.

2.
If \(\Gamma \) and \(\Gamma '\) are coedgeregular graphs with parameters \((v,k,\mu )\) and \((v',k',\mu ')\) where \(vv' = k  k' = \mu  \mu '\), then their complete union is coedgeregular with parameters \((v + v', k + v', \mu + v').\)
Definition 2.4
Let \(\Gamma =(V,E)\) be an edgeregular graph with parameters \((v,d,\lambda )\), \(D=(V,{\mathcal {B}})\) is a minimal \((v,k,\lambda )\)design over \(\Gamma \) (i.e., having least number of blocks) if \(\{i,j\}\in E\), then \(\{i,j\}\) is contained in exactly \(\lambda \) blocks, where \({\mathcal {B}}\) is a set of maximum cliques over \(\Gamma \).
Example 2.5
\(\Gamma =L_2(4)\), and maximum clique is all the rows (columns) of the array. \({\mathcal {B}}=\{ \{1,2,3,4\} , \{5,6,7,8 \}, \{9,10,11,12\}, \{13,14,15,16\}\}\). Here, if \(\{i,j\}\in E\), then \(\{i,j\}\) is in exactly \(\lambda =1\) blocks.
We will need the following classification results of [3] and [8] on connected regular graphs with least eigenvalue \(2\).
Theorem 2.6
([2], Theorem 3.11.4) Let \(\Gamma \) be a connected regular graph with smallest eigenvalue \(2\). Then,

1.
(Seidel [8]) If \(\Gamma \) is strongly regular, then \(\Gamma \) is a triangular graph \(T(n) (n \ge 5)\), a square grid \(n \times n\) (also called a lattice graph \(L_2 (n)(n \ge 3),\) a complete multipartite graph \(K_{n\times 2} (n \ge 2)\) or one of the graphs of Petersen, Clebsch, Schläfli, Shrikhande or Chang.

2.
If \(\;\Gamma \) is edgeregular, then \(\Gamma \) is strongly regular or the line graph of a regular trianglefree graph.

3.
If \(\;\Gamma \) is amply regular, then \(\Gamma \) is strongly regular or the line graph of a regular graph of girth \(\ge 5.\)

4.
If \(\;\Gamma \) is coedgeregular, then \(\Gamma \) is strongly regular, an \(m \times n\)grid or one of the two regular subgraphs of the Clebsch graph with 8 and 12 vertices, respectively.
Theorem 2.7
([9], Theorem 4.1.) Let \(\Gamma \) be an edgeregular graph with parameters \((v, d, \lambda ),\) having at least one edge. Then, \(\Gamma \) is a complete multipartite if and only if \(v = 2d \lambda \).
3 Classification of minimal \((v, k, \lambda ,\mu )\)designs over strongly regular graphs
Table 1 gives classification obtained earlier of two class partially balanced \((v, k, \lambda ,\mu )\)designs with replication number \(r=2\lambda \mu \), associated with strongly regular graphs with smallest eigenvalue \(2\). Here, \([n]=\{1, 2, \ldots , n\}; n\ge 4\).
4 Designs associated with edgeregular graphs
Definition 4.1
Let \(\Gamma =(V,E)\) be an edgeregular graph parameters \((v,d,\lambda )\), where \(\lambda \ge 1\). A \((v,k,\lambda ,\zeta )\)design over \(\Gamma \) is a pair \(D=(V,{\mathcal {B}})\), where \({\mathcal {B}}\) is a set of ksubsets of V, \(2\le k \le v1\), called blocks, satisfying the following conditions:

1.
\(B=k\) for each \(B\in {\mathcal {B}}\);

2.
if \(i,j\in V, i\ne j\) and \(\{i,j\}\in E\), then \(\{i,j\}\) is contained in exactly \(\lambda \) blocks \(B\in {\mathcal {B}}\) and if \(\{i,j\}\notin E\), then \(\{i,j\}\) is contained in exactly \(\zeta \) blocks \(B\in {\mathcal {B}}\).
If \(\lambda =\zeta \), then design D is a 2\((v,k,\lambda )\) design. So we assume \(\lambda \ne \zeta \) to distinguish from this type of design.
Proposition 4.2
Let \(D=(V,{\mathcal {B}})\) be a \((v,k,\lambda ,\zeta )\)design over an edgeregular graph \(\Gamma \) with parameters \((v,d,\lambda )\), where \(\lambda \ge 1\). Then, there is an integer r (the replication number of D), such that any point \(i\in V\) is contained in exactly r blocks. Further the following relations hold:
where \(b={\mathcal {B}}\).
Proof
Fix \(i\in V\) and count in two ways pairs (j, B), where \(j\in V, j\ne i, B\in {\mathcal {B}}\) and \(\{i,j\} \subseteq B\). We then obtain (4.1). Next, count in two ways pairs (i, B) , where \(i \in V\) and \(i\in B\in {\mathcal {B}}\), to get (4.2). \(\square \)
Theorem 4.3
Let \(\Gamma =(V,E)\) be an edgeregular graph with parameters \((v,d,\lambda )\), where \(\lambda \ge 1\), and let \(D=(V,{\mathcal {B}})\) be a \((v,k,\lambda ,\zeta )\)design over \(\Gamma \), with b and r being the number of blocks and replication number of D, respectively. Let A be the adjacency matrix of \(\Gamma \) and \({\overline{A}}=JIA\), the adjacency matrix of the complement of \(\Gamma \). If \(\dfrac{r\lambda }{\lambda \zeta }\) is not a multiple eigenvalue of \({\overline{A}}\), then \(b\ge v\).
Proof
As earlier, take \(V=\{1,2,\ldots ,v\}\) and associate with each \(B\in {\mathcal {B}}\), a variable \(x_B\). Let \(x_B\) act on V by
Let \({\mathcal {P}}\) be a vector space (over \({\mathbb {Q}}\)) of linear polynomials in variables \(x_B, B\in {\mathcal {B}}\). Clearly, \(\dim ({\mathcal {P}})=b+1\). For \(i=1,2,\ldots ,v\), define polynomial \(f_i\), by
Then, for \(i,j \in V\),
Claim 1: At least \(v1\) of the polynomials \(f_i\) are linearly independent.
If all v of the polynomials \(f_i\) are linearly independent, then clearly the claim follows. So assume \(\sum _{i=1}^{v} \alpha _i f_i={\overline{0}}\) is a dependency relation for some \(\alpha _1,\alpha _2,\ldots ,\alpha _v \in {\mathbb {Q}}\), not all zero.
Let \(j\in V\), then \(\sum _{i=1}^{v} \alpha _i f_i(j) =0\), so
Put \({\overline{a}}=(\alpha _1,\alpha _2, \ldots , \alpha _v)^t\). Then, using (4.3), we get
i.e., \((JIA){\overline{a}} = \left( \dfrac{r\lambda }{\lambda \zeta }\right) {\overline{a}}\), and \({\overline{a}}\ne {\overline{0}}.\) So \(\dfrac{r\lambda }{\lambda \zeta }\) is an eigenvalue of \({\overline{A}}\), with eigenvector \({\overline{a}}\).
Now, by assumption the eigenspace of \(\dfrac{r\lambda }{\lambda \zeta }\) is one dimensional. Let \({\overline{\beta }}=(\beta _1,\beta _2,\ldots ,\beta _v)^t\) span the eigenspace, so \({\overline{a}}=c{\overline{\beta }}\), for some \(c\in {\mathbb {Q}}\). Without loss of generality, assume that \(\beta _1\ne 0\). Since we are assuming \(\sum _{i=1}^{v} \alpha _i f_i={\overline{0}}\) is a dependency relation, we get
This implies \(\sum _{i=1}^{v} \beta _i f_i={\overline{0}}.\) Since \({\overline{b}}\ne {\overline{0}}\), without loss of generality assume that \(\beta _1\ne 0\), so \(f_1= \sum _{i=1}^{v} \dfrac{\beta _i}{\beta _1} f_i\). Assume if possible that \(\sum _{i=1}^{v} \gamma _i f_i={\overline{0}}\) is a dependence relation. Then, \(0f_1+ \sum _{i=2}^{v} \gamma _i f_i={\overline{0}}\) is a dependence relation between \(f_1,f_2,\ldots ,f_v\). This implies that \({\overline{u}}=(0,\gamma _2,\gamma _3,\ldots ,\gamma _v)\) is an eigenvector of \({\overline{A}}\) corresponding to the eigenvalue \(\dfrac{r\lambda }{\lambda \zeta }\). This implies that \((0,\gamma _2,\gamma _3,\ldots ,\gamma _v)=c(\beta _1,\beta _2,\ldots ,\beta _v)\), for some \(c\in {\mathbb {Q}}, c\ne 0.\) This implies \(c\beta _1=0\), a contradiction, since \(\beta _1\ne 0\). Thus, \(f_1,f_2,\ldots ,f_v\) span a subspace of \({\mathcal {P}}\) of \(\dim \ge v1\).
Claim 2: \(\sum _{B\in {\mathcal {B}}}^{} x_B\) is not in the linear span of \(f_1,f_2,\ldots ,f_v\).
Suppose \(\sum _{B\in {\mathcal {B}}}^{} x_B=\sum _{i=1}^{v} \alpha _i f_i\) for some \(\alpha _1,\alpha _2,\ldots ,\alpha _v\in {\mathbb {Q}}\). Then,
Equating constant term on both sides of (4.4), we get \(\sum _{i=1}^{v} \alpha _i \lambda =0\), which implies \(\sum _{i=1}^{v} \alpha _i =0\), since \(\lambda \ne 0\).
Next, applying (4.4) to a point \(j\in V\), we obtain
Summing up (4.5) for \(j=1,2,\ldots ,v\), we get
a contradiction. Thus, \(\sum _{B\in {\mathcal {B}}}^{} x_B\) is not in the linear span of \(f_1,f_2,\ldots ,f_v\).
This implies that \(b+1=\dim ({\mathcal {P}})\ge v\), which gives \(b\ge v1\). Now, if \(b=v1\), then \(vr=bk\) gives \(vr=(v1)k\), which then implies v divides k, and then, \(k\ge v\), a contradiction. This finally shows that \(b\ge v\). \(\square \)
The following lemma is just the statement of Result 1.1, part 2.
Lemma 4.4
Let \(\Gamma =(V,E)\) be an edgeregular graph with parameters \((v,d,\lambda )\), where \(\lambda \ge 1\), and let \(D=(V,{\mathcal {B}})\) be a \((v,k,\lambda ,\zeta )\)design over \(\Gamma \), with r being the replication number of D. Then,

1.
\(r\ge 2 \lambda \zeta \);

2.
If \(r= 2 \lambda \zeta \), then for any three vertices \(x,y,z\in V\), such that \(\{x,y\}, \{x,z\}\in E\), and \((y,z)\notin E\),

(a)
any block containing x must contain y or z, and

(b)
any block containing y and z must contain x.

(a)
Remark 4.5

1.
Let \(\Gamma \) be an edgeregular graph with parameters \((v,d,\lambda )\). The number of blocks b of a \((v,k,\lambda ,\zeta )\)design (\(V,{\mathcal {B}})\) over \(\Gamma \) is given by
$$\begin{aligned} b=\dfrac{v r^2}{(v1d)\zeta +d\lambda +r} \end{aligned}$$and is an increasing function of r.

2.
Hence, designs with \(r=2\lambda \) have the minimum number of blocks. If this minimum number is less than v, then by Theorem 4.3, \(\dfrac{r\lambda }{\lambda \zeta }\) is a multiple eigenvalue of \({\overline{A}}=JIA\), the adjacency matrix of the complement of \(\Gamma \).

3.
\({\mathcal {B}}\) is the set of cliques of \(\Gamma \) if and only if \(\zeta =0\).

4.
Let \(\Gamma \) be strongly regular \((v,d,\lambda ,\mu )\) graph with minimum eigenvalue \(m\) and \(D=(V,{\mathcal {B}})\) be a \((v,k,\lambda ,\zeta )\)design over \(\Gamma \). If \(\zeta =0\), then by Proposition 2.2, we get \(k\le 1+d/m\) because every block is a clique of \(\Gamma \).
Theorem 4.6
Let G be a regular trianglefree graph with valency k and \(\Gamma =(V,E)=L(G)\) be the line graph of G. Then, \(\Gamma \) is an edgeregular \((v,d,\lambda )=(Ek/2,2(k1), k2)\) graph. Let \(D=(V,{\mathcal {B}})\) be a \((v,k,\lambda ,\zeta )\)design over \(\Gamma \), where \({\mathcal {B}}\) is a set of maximum cliques in \(\Gamma \). Moreover, \(r=2, d=2(k1), \zeta =0, {\mathcal {B}}=V(G), \lambda =1\) and \(k=3\).
Proof
As G is trianglefree, a maximum clique of \(\Gamma \) contains a set of edges in G incident on a vertex. So, size of a maximum clique in \(\Gamma \) is k, the valency of G. So block size of D is k and \({\mathcal {B}}=V(G)\). If \(\{i,j\}\in E\), then i and j are edges in G incident on a vertex, so \(\{i,j\}\) is contained in exactly \(\lambda =1\) block. But in G there are \(k2\) edges other than i and j incident on x. Hence, \(\lambda =k2\). So we have \(k=3\). If \(i\in V\), then \(i\in E(G)\) and i is part of two set of edges incident on end vertices of i. So \(r=2\). \(\square \)
Theorem 4.7
Let G be a regular trianglefree graph with valency \(\ell \ge 3\) and \(\Gamma =L(G)\) be the line graph of G. Then, \(\Gamma \) is an edgeregular \((v,d,\lambda )=(V(G)\ell /2,2(\ell 1), \ell 2)\) graph. Let \(D=(E(G),{\mathcal {B}})\) be a \((v,k,\lambda ,\zeta )\)design over \(\Gamma \), where \({\mathcal {B}}\) is a set of 3subsets maximum cliques in \(\Gamma \). Moreover, \(r=(\ell 1)(\ell 2),\zeta =0, {\mathcal {B}}=V(G)\times \genfrac(){0.0pt}0{\ell }{3}\) and \(k=3\).
Proof
As blocks are subsets of maximum cliques of \(\Gamma \), we have \(\zeta =0\). Since G is a regular trianglefree graph of degree \(\ell \), \(\Gamma \) is a regular graph of degree \(d=2(\ell 1)\).
Any two adjacent vertices in \(\Gamma \) belong to exactly one maximum clique of \(\Gamma \). The size of each maximum clique in \(\Gamma \) is \(\ell \), and two distinct maximum cliques in \(\Gamma \) have exactly one element in common. There are V(G) maximum cliques in \(\Gamma \).
As blocks are 3 subsets of maximum cliques in \(\Gamma \), \(\lambda =\ell 2\). Each vertex of \(\Gamma \) belongs to two maximum cliques in \(\Gamma \), and hence, \(r=2 \genfrac(){0.0pt}0{\ell 1}{2}=(\ell 1)(\ell 2)\) and \({\mathcal {B}}=V(G) \times \genfrac(){0.0pt}0{\ell }{3}\). \(\square \)
Theorem 4.8
If \(\Gamma =K_{n\times 2}=\overline{n K_2}, n\ge 3\), then \(\Gamma \) is an edgeregular \((2n, 2(n1), 2(n2))\)graph. Let \((V(\Gamma ),{\mathcal {B}})\) be a \((v,k,\lambda ,\zeta )\)design over \(\Gamma \), where \({\mathcal {B}}\) is a set of maximum cliques of \(\Gamma \). Then, \(n=3,4\).
Proof
Let \(V(\Gamma )=\{a_i,b_i i=1,2,\ldots ,n\}\) with \(\{a_i , a_j\},\{b_i , b_j\},\{a_i , b_j\}\) are edges in \(\Gamma \), for \(1\le i,j\le n\), if and only if \(i\ne j\). If B is a maximum clique in \(\Gamma \), then \(V(B)=\{x_i i=1,2,\ldots , n\}\), where \(x_i\) is either \(a_i\) or \(b_i\). Hence, if \({\mathcal {B}}'\) is the set of all maximum cliques of \(\Gamma \), then \({\mathcal {B}}'=2^n\) and \(B=n\), for all \(B\in {\mathcal {B}}'\). Observe that every \(v\in V(\Gamma )\) is contained in \(2^{n1}\) maximum cliques in \(\Gamma \).
Then, \((V(\Gamma ),{\mathcal {B}})\) is a \((2n,n, 2(n2),0)\)design, with parameters \(b=8(n2)\) and \(r=4(n2)\). If \({\mathcal {B}}={\mathcal {B}}'\), then \(8(n2)=2^n\), and hence, \(n=3,4\).
Suppose \((V(\Gamma ),{\mathcal {B}})\) exists for \(n\ge 5\). Consider \({\mathcal {B}}''={\mathcal {B}}' {\mathcal {B}}\), then \({\mathcal {B}}''=2^n8(n2)\) and each vertex of \(\Gamma \) is contained in \(2^{n1}4(n2)\) maximum cliques in \({\mathcal {B}}''\). Count the pairs (x, B), where \(x\in V(\Gamma )\) and \(B\in {\mathcal {B}}''\), in two different ways, to see that
This implies \(n\le 8\) as \(2^n\) divides \(8(n2)n\). From (4.6), we conclude that \(n\le 4\). \(\square \)
Remark 4.9
Designs obtained in Theorems 4.6 and 4.8 are minimal designs in the sense that \(r=2\lambda \zeta , \zeta =0\) over a graph with smallest eigenvalue \(2\).
Remark 4.10
Designs obtained in Theorem 4.7 are minimal in the sense that \(\zeta =0\) and designs are over graphs with smallest eigenvalue \(2\). Further, observe that \(r=2\lambda \) if and only if \(\ell =3\).
Theorem 4.11
Let \(\Gamma \) be an edgeregular graph with parameters \((v, d, \lambda ),\) having at least one edge, with \(v=2d\lambda \). If k divides \(2 \lambda (\lambda +4)\), then there is a \((v,k,\lambda ,\zeta )\) minimal design over \(\Gamma \) in the sense that \(r=2\lambda \) and \(\zeta =0\), whose blocks are maximum cliques of size k and \(d=2(k1)\).
Proof
By Theorem 2.7, \(\Gamma \) is a complete multipartite graph, with say k parts. Then, the size of maximum cliques of \(\Gamma \) is k. If \({\mathcal {B}}\) is the set of all maximum cliques of \(\Gamma \), then \(D=(V,{\mathcal {B}})\) is a design over \(\Gamma \). Using equation (4.1), we get \(d=2(k1)\), and using equation (4.2), we get \(b= 8 \lambda \dfrac{2 \lambda (\lambda +4)}{k}\). Hence, \(v=4 (k1)\lambda \). \(\square \)
5 Examples of designs over strongly regular graphs considered as edgeregular graphs
In this section, we try to find examples of designs over strongly regular graph with smallest eigenvalue \(2\) and \(r=2 \lambda \zeta \). If \(\zeta =0\), then we explore the possibility for blocks as cliques of \(\Gamma \). We use Eq. (4.2) to see that \(vk=\dfrac{(2 kd2) (\lambda \zeta )}{\zeta }\). Hence, we take \(1\le \zeta < \lambda \). Other parameters are calculated with the help of Eqs. (4.1) and (4.2). We list our findings in Table 2.
6 Possible parameters of designs on edgeregular graphs
In this section, we give possible parameters of designs associated with edgeregular graphs \(\Gamma \), which are not strongly regular, with parameters \((v,d,\lambda ), \lambda \ge 1\), with smallest eigenvalue \(2\) and \(r=2 \lambda \zeta \). As before, we consider \(0\le \zeta < \lambda \). We refer to [2] for details of graphs listed in this section.
Example 6.1
Possible parameters of minimal designs over edgeregular graphs.

(a)
Hall–Janko near octagon \((v,d,\lambda )= (315,10,1),\) design parameters \(v=315, b=105, k =6, r=2, \zeta =0.\)

(b)
Conway–Smith graph, \((v,d, \lambda )=(63,10,3),\) design parameters \(v=63, b=63, r=6=k, \zeta =0.\)

(c)
Icosidodecohedral graph \((v,d,\lambda )=(30,4,1),\) design parameters \(v=30, b=20, r=2, k=3, \zeta =0.\)

(d)
From A. A. Makhnev graph \((v,d, \lambda )=(64,52,42),\) design parameters \(v=64, b=148, r=74, k=32, \zeta =10,\) (See Makhnev, [7]).
7 Designs associated with coedgeregular graphs
In this section, we consider a graph \(\Gamma \) satisfying R3 with \(\mu \) positive. Then, \(\Gamma \) is connected with diameter at most 2.
Definition 7.1
Let \(\Gamma =(V,E)\) be a coedgeregular graph with parameters \((v,d,\mu )\), where \(\mu \ge 1\). A \((v,k,\mu ,\xi )\)design over \(\Gamma \) is a pair \(D=(V,{\mathcal {B}})\) such that \({\mathcal {B}}\) is a set of ksubsets of V, \(2\le k \le v1\), called blocks, satisfying the following conditions:

1.
\(B=k\) for each \(B\in {\mathcal {B}}\);

2.
if \(i,j\in V, i\ne j\) and \(\{i,j\}\notin E\), then \(\{i,j\}\) is contained in exactly \(\mu \) blocks \(B\in {\mathcal {B}}\), and if \(\{i,j\}\in E\), then \(\{i,j\}\) is contained in exactly \(\xi \) blocks \(B\in {\mathcal {B}}\).
If \(\mu =\xi \), then design D is a 2\((v,k,\lambda )\) design. So we assume \(\mu \ne \xi \) to distinguish from this type of design.
Proposition 7.2
Let \(D=(V, {\mathcal {B}})\) be a \((v,k,\mu ,\xi )\)design over a coedgeregular graph \(\Gamma =(V,E)\) with parameters \((v,d,\mu )\), where \(\mu \ge 1\). Then, there is an integer r (the replication number of D) such that any point \(i\in V\) is contained in exactly r blocks. Further the following relations hold:
where \(b={\mathcal {B}}\).
Proof
We obtain (7.1) by fixing \(i\in V\), and counting in two ways pairs (j, B), where \(j\in V, j\ne i,\) and \(B\in {\mathcal {B}}\) with \(\{i,j\} \subseteq B\). By counting in two ways pairs (j, B), where \(B\in {\mathcal {B}}\), and \(j\in B\), we obtain (7.2). \(\square \)
We next state the following theorem, without proof as it is similar to that for edgeregular case.
Theorem 7.3
Let \(\Gamma =(V,E)\) be a coedgeregular graph with parameters \((v,d,\mu )\), where \(\mu \ge 1\), and let \(D=(V,{\mathcal {B}})\) be a \((v,k,\mu ,\xi )\)design over \(\Gamma \), with b and r being the number of blocks and replication number of D, respectively. Let A be the adjacency matrix of \(\Gamma \). If \(\dfrac{r\mu }{\mu \xi }\) is not a multiple eigenvalue of \({\overline{A}}=JIA\), the adjacency matrix of the complement of \(\Gamma \), then \(b\ge v\).
The proof of the next lemma is also clear.
Lemma 7.4
Let \(D=(V,{\mathcal {B}})\) be a \((v,k,\mu ,\xi )\)design over a coedgeregular graph \(\Gamma =(V,E)\) with parameters \((v,d,\mu )\), where \(\mu \ge 1\) and r be the replication number of D. Then,

1.
\(r\ge 2 \mu \xi \);

2.
If \(r= 2 \mu \xi \), then for any three vertices \(x,y,z\in V\), such that \(\{x,y\}, \{x,z\}\notin E\), and \((y,z)\in E\),

(a)
any block containing x must contain y or z, and

(b)
any block containing y and z must contain x.

(a)
We can make the following observations concerning a \((v,k,\mu ,\xi )\)design over a coedgeregular graph \(\Gamma \) with parameters \((v,d,\mu )\).
Remark 7.5

1.
Let \(\Gamma \) be a coedgeregular graph with parameters \((v,d,\mu )\). The number of blocks b of a \((v,k,\mu ,\xi )\)design \((V,{\mathcal {B}})\) over \(\Gamma \) is given by
$$\begin{aligned} b=\dfrac{v r^2}{r+d\xi +(vd1)\mu } \end{aligned}$$and is an increasing function of r.

2.
Hence, designs with \(r=2\mu \xi \) have the minimum number of blocks. If this minimum number is less than v, then by Theorem 7.3, \(\dfrac{r\mu }{\mu \xi }=1\) is a multiple eigenvalue of \({\overline{A}}=JIA\), the adjacency matrix of the complement of \(\Gamma \). (Equivalently \(\dfrac{r\mu }{\mu \xi }=2\) is a multiple eigenvalue of A.) This explains our interest in coedgeregular graphs with least eigenvalue \(2\).

3.
\({\mathcal {B}}\) is the set of cocliques of \(\Gamma \) if and only if \(\xi =0\).

4.
Let \(\Gamma \) be a regular graph with v vertices, valency d, and with smallest eigenvalue \(m\) and \(D=(V,{\mathcal {B}})\) be a \((v,k,\mu ,\xi )\)design over \(\Gamma \). If \(\xi =0\), then as blocks are coclique of \(\Gamma \), by Proposition 2.2, we get \(k \le (1+d/m)^{1}v\).
Proposition 7.6
Let \(D=(V,{\mathcal {B}})\), where \({\mathcal {B}}\) is the sets of maximum cocliques of \(\Gamma \), be a \((v,k,\mu ,\xi )\)design over a coedgeregular graph \(\Gamma \) with parameters \((v,d,\mu )\). Then, D is a \((v,k,\mu ,\xi )\)design with replication number \(r=\mu \) if and only if \(\Gamma \) is isomorphic to the complete bipartite graph \(K_{d,d}\), and then, \(v=2d, k=d, \xi =0,\) and \(\mu =1\).
Proof
Suppose \(D=(V,{\mathcal {B}})\) is a \((v,k,\mu )\)design over \(\Gamma \), where \({\mathcal {B}}\) is the sets of maximum cocliques of \(\Gamma \). Suppose \(r=\mu \), then \(\xi =0\) and the relation \((vd1)\mu =\mu (k1)\) gives \(k=vd\). Let S be a set of maximum cocliques in \(\Gamma \), and hence, \(S = v  d.\) Now, we claim that \(v  d = d.\) For, if \(v  d > d\), then degree of a vertex u in \(V  S\) is greater than d since every vertex in S is adjacent to every vertex in \(V  S,\) which contradicts the valency of \(\Gamma \). On the other hand, if \(v  d < d,\) then \(\Gamma \) contains only one maximum coclique, namely S itself, which implies that a vertex in \(V  S\) does not belong to any maximum coclique, a contradiction to the fact that D is a \((v,k,\mu ,\xi )\)design. This implies that \(v = 2d\) and \(k = d.\) This, in turn, implies that \(G = K_{ d,d}\) and hence \(\mu = 1.\) \(\square \)
8 Example of designs over strongly regular graphs considered as coedgeregular graphs
In this section, we try to find examples of designs over coedgeregular graph with smallest eigenvalue \(2\) and \(r= 2 \mu \xi \). If \(\xi =0\), then we explore the possibility for blocks as cocliques of \(\Gamma \). We use equation (7.1) to see that \(vk=\frac{(d+k1) (\mu \xi )}{\mu }\). Hence, we consider \(0\le \xi < \mu \). Other parameters are calculated with the help of equations (7.1) and (7.2). We list our findings in Tables 3, 4 and 5.
Example 8.1
Let \(\Gamma \) be a \(m \times n\)grid graph, with \(n\ge m\). Then, \(\Gamma \) is a coedgeregular graph with parameters \((mn,m+n2,2)\).
For \(\xi =0\), observe that \(b12=\dfrac{4 (m3) (n3)}{(m1) (n1)+2}\) and \(k=m+\dfrac{1}{2} (m1) (n3)\). As blocks are cocliques of \(\Gamma , k\le m\) and \(\Gamma \) has \(n(n1)\cdots (nm+1)\) cocliques of size m. So \(n\le 3\). We list the possible values of other parameters in Table 3.
For \(\xi =1\), observe that \(b=\dfrac{9 m n}{2 n mmn+3}, r=3\) and \(k=\dfrac{1}{3} (2 n mmn+3)\). As \(\dfrac{ m n}{2 n mmn+3}<1\), \(b\le 8\). We write \(m=\dfrac{b (n3)}{(2 n 1)b9 n}\) and observe that \(m=1\). We list the possible values of other parameters in Table 3.
9 Possible parameters of designs associated with coedgeregular graphs
10 Designs associated with amply regular graphs
In this section, we assume that \(\Gamma \) to be a connected amply regular graph.
Lemma 10.1
([2], Lemma 1.1.1) In any amply regular graph \(\Gamma \) with parameters \((v,d,\lambda ,\mu )\), the number of points \(k_2\) at distance 2 from any vertex \(\gamma \) is independent of \(\gamma \) and satisfies the relation
Proof
Count the number of edges between \(\Gamma (\gamma )\) and \(\Gamma _2(\gamma )\) in two ways. \(\square \)
Definition 10.2
Let \(\Gamma =(V,E)\) be an amply regular graph with parameters \((v,d,\lambda ,\mu )\), where \(\lambda \ge 1, \mu \ge 1,\) and \(\lambda \ne \mu \). A \((v,k,\lambda ,\mu ,\xi )\)design on \(\Gamma \) is a pair \(D=(V,{\mathcal {B}})\), where \({\mathcal {B}}\) is a set of ksubsets of V, \(2\le k \le v1\), called blocks, satisfying the following conditions:

1.
if \(i,j\in V\), \(i\ne j\), and \(\{i,j\}\in E\), then there are exactly \(\lambda \) blocks containing \(\{i,j\},\)

2.
if \(i,j\in V\), \(i\ne j\), such that \(d(i,j)=2\), then there are exactly \(\mu \) blocks containing \(\{i,j\},\)

3.
if \(i,j\in V\), \(i\ne j\), such that \(d(i,j)>2\), then there are exactly \(\xi \) blocks containing \(\{i,j\}.\)
Clearly, \(\mu \ge 1\). If \(\lambda =\mu =\xi >0\), then design D is a 2\((v,k,\lambda )\) design. So we assume \(\lambda \ne \mu \) or \(\lambda \ne \xi \) to distinguish from this type of design.
Lemma 10.3
Let \(D=(V, {\mathcal {B}})\) be a \((v,k,\lambda ,\mu ,\xi )\)design over an amply regular graph with parameters \((v,d,\lambda ,\mu )\), where \(\lambda \ge 1, \mu \ge 1\), and \(\lambda \ne \mu \). Then, there is an integer r (the replication number of D) such that any point \(i\in V\) is contained in exactly r blocks and
where \(b={\mathcal {B}}\).
Proof
We obtain (10.2) by fixing \(i\in V\) and counting in two ways, pairs (j, B), where \(j\in V\), \(i\ne j\), \(B\in {\mathcal {B}}\) and \(\{i,j\}\subset B\). By counting in two ways pairs (j, B), where \(B\in {\mathcal {B}}\), and \(j\in B\), we obtain (10.3). \(\square \)
Theorem 10.4
Let \(\Gamma =(V,E)\) be an amply regular graph, with parameters \((v,d,\lambda ,\mu )\), where \(\lambda \ge 1, \mu \ge 1\), \(\lambda \ne \mu \), and let \(D=(V, {\mathcal {B}})\) be a \((v,k,\lambda ,\mu ,\xi )\)design over \(\Gamma \), with b and r being the number of blocks and replication number of D, respectively. Let A be the adjacency matrix of \(\Gamma \) and \(A'=(\lambda \xi  (\mu \xi ) \mu \lambda ) A + (\mu \xi ) \mu A^2\). If \((d\mu (\mu \xi )(r\xi ))\) is not a multiple eigenvalue of \(A'\), then \(b\ge v\).
Proof
We assume that \(V=\{1,2,\ldots , v\}\), with each block \(B\in {\mathcal {B}},\) we associate variable \(x_B\). We define the action of these variables in \(i\in V\) by
Let \({\mathcal {P}}\) be a vector space (over \({\mathbb {Q}}\)) of linear polynomials in variables \(x_B, B\in {\mathcal {B}}\). Clearly, \(\dim ({\mathcal {P}})=b+1\). For \(i=1,2,\ldots ,v\), define polynomial
Then, for \(i,j \in V\),
We claim that at least \(v1\) of the polynomials \(f_i, 1\le i \le v\) are linearly independent. This is clearly true if \(f_i, 1\le i \le v\) are linearly independent. Suppose \(\sum _{i=1}^{v} \alpha _i f_i =0\) \((\alpha _i \in {\mathbb {Q}})\) is dependence relation.
Let \(j\in V\), then \(\sum _{i=1}^{v} \alpha _i f_i(j) =0\), so
Let \({\overline{a}}=(\alpha _1,\alpha _2, \ldots , \alpha _v)\). Then, we can rewrite (10.4) as
which gives
Since by assumption, \({\overline{a}}\ne {\overline{0}}\), this implies \((d\mu (\mu \xi )(r\xi ))\) is an eigenvalue of \(A'\) with eigenvector \({\overline{a}}\). By hypothesis, the multiplicity of this eigenvalue is 1.
Let \({\overline{b}}=(\beta _1,\beta _2,\ldots ,\beta _v)\) be the basis vector of the eigenspace corresponding to the eigenvalue \((d\mu (\mu \xi )(r\xi ))\). Then, \({\overline{a}}=c {\overline{b}}\), for some \(c\in {\mathbb {Q}}, c\ne 0\). Then, \(\sum _{i=1}^{v}\beta _i f_i=0\) is also a dependence relation.
Without loss of generality assume that \(\beta _1\ne 0\). Then, \(f_1=\sum _{i=2}^{v} \dfrac{\beta _i}{\beta _1}f_i\). Assume if possible that \(\sum _{i=2}^{v} \gamma _i f_i=0\) is a dependence relation. Then, \(0f_1+\sum _{i=2}^{v} \gamma _i f_i=0\) is dependence relation between \(f_1,f_2,\ldots ,f_v\). This implies that \({\overline{u}}=(0,\gamma _2,\gamma _3,\ldots ,\gamma _v)\) is an eigenvector of \(A'\) corresponding to the eigenvalue \((d\mu (\mu \xi )(r\xi ))\).
This means that \((0,\gamma _2,\gamma _3,\ldots ,\gamma _v)=c(\beta _1,\beta _2,\ldots ,\beta _v)\) for some \(c\ne 0\), a contradiction. Thus, \(f_1,f_2,\ldots ,f_v\) span a subspace of \({\mathcal {P}}\) of dimension \(\ge v1\).
We claim that the polynomial \(\sum _{B \in {\mathcal {B}}}^{} x_B \) is not in the linear span of \(f_1,f_2,\ldots ,f_v\).
Suppose \(\sum _{B \in {\mathcal {B}}}^{} x_B = \sum _{i=1}^{v} \alpha _i f_i\), for some \(\alpha _1,\alpha _2,\ldots ,\alpha _v \in {\mathbb {Q}}\). This implies
Then, equating the constant term on both sides of (10.5) of the point \(j\in V\), we obtain
Summing (10.6) for \(j=1,2, \ldots ,v\), gives
Since \(\sum _{j=1}^{v} \alpha _j=0\), this implies \(vr=0\), a contradiction.
Thus, \(\sum _{B \in {\mathcal {B}}}^{} x_B\) is not in the linear span of \(f_1,f_2,\ldots , f_v\). Since \(\dim {\mathcal {P}}=b+1\), this means that \(b\ge v1\). Now, if \(b=v1\), then (10.3) implies that \(v1\) divides r, so \(r=v1\) and \(k=v1\), a contradiction. Thus, \(b\ge v\). \(\square \)
Lemma 10.5
Let \(D=(V,{\mathcal {B}})\) be a \((v,k,\lambda ,\mu ,\xi )\)design over an amply regular graph \(\Gamma =(V,E)\), with parameters \((v,d,\lambda ,\mu )\), where \(\lambda \ge 1, \mu \ge 1\), and r be the replication number of D. Then,

1.
\(r\ge 2 \lambda \mu \);

2.
If \(r= 2 \lambda \mu \), then for any three vertices \(x,y,z\in V\), such that \(\{x,y\}, \{x,z\}\in E\), and \(d(y,z)=2\),

(a)
any block containing x must contain y or z, and

(b)
any block containing y and z must contain x.

(a)
Proof
Let x, y, z be three distinct vertices such that \(\{x,y\}, \{x,z\}\in E\), and \(d(y,z)=2\). There are r blocks containing x, \(\lambda \) blocks containing \(\{x,y\}\) and \(\lambda \) blocks containing \(\{x,z\}\). Let \(\alpha \) be the number of blocks containing \(\{x,y,z\}\). Then, there are \(\lambda \alpha \) blocks, containing \(\{x,y\},\) but not z and \(r\lambda \) blocks, containing x but not z. Therefore, \(r\lambda \ge \lambda \alpha \) which implies \(r\ge 2 \lambda \alpha \). Since \(d(y,z)=2\), there are exactly \(\mu \) blocks containing \(\{y,z\}\), so \(\mu \ge \alpha \). This proves \(r\ge 2 \lambda  \mu \).
Now, suppose \(r=2 \lambda \mu \). Then, \(r\lambda =\lambda \mu \ge \lambda \alpha \) and \(\alpha =\mu \). Therefore, every block containing x but not z must contain y, proving 2 (a), and every block containing y and z must contain x, proving 2 (b). \(\square \)
Remark 10.6

1.
Let \(\Gamma \) be an amply regular graph with parameters \((v,d,\lambda ,\mu )\). The number of blocks b of a \((v,k,\lambda ,\mu ,\xi )\)design \(D=(V,{\mathcal {B}})\) over \(\Gamma \) is given by
$$\begin{aligned} b=\dfrac{v r^2}{d\lambda +k_2\mu + (v1dk_2) \xi +r} \end{aligned}$$and is an increasing function of r.

2.
Hence, designs with \(r=2\lambda \mu \) have the minimum number of blocks. If this minimum number is less than v, then by Theorem 10.4, \((d\mu (\mu \xi )(r\xi ))\) is a multiple eigenvalue of \(A'\), which is equivalent to \(\dfrac{r\mu }{\mu \lambda }=2\). This may explain our interest in \((v,k,\lambda ,\mu ,\xi )\)designs over amply regular graphs with an eigenvalue \(2\).
11 Concluding remarks
In the present paper, we have extended the notion of designs over regular graphs, developed by Ionin and Shrikhande [5], to designs over edgeregular graphs, amply regular graphs and coedgeregular graphs. We have also obtained partial classification of minimal designs with smallest eigenvalue \(2\) in all cases. The emphasis was given on construction of minimal designs over nonstrongly regular graphs. In the case of amply regular graphs, construction of designs is challenging. The combination of cliques and cocliques may obtain examples of these types of designs. To find cliques or cocliques of a graph and also to perform other graph operations, we have used computer algebra system SageMath [10].
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Shrikhande, M.S., Pawale, R.M. & Yadav, A.K. Designs over regular graphs with least eigenvalue \(2\). J Algebr Comb 54, 1021–1045 (2021). https://doi.org/10.1007/s10801021010368
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DOI: https://doi.org/10.1007/s10801021010368
Keywords
 Edgeregular graph
 Amply regular graph
 Coedgeregular graph
 Strongly regular graph
 Least eigenvalue \(2\)
 Hoffman bound
 Partially balanced design