## 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 $$2\le k \le v-1$$ 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 k-subsets 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 2-class 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. 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. 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 $$\{y,z\} \notin E$$,

1. (a)

any block containing x must contain y or z and

2. (a)

any block containing y and z must contain x.

3. 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. 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 point-block pair (xB), 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. 1.

the complete n-partite graph $$K_{n\times 2}$$, with parameters $$(2n,2n-2,2n-4,2n-2), n \ge 2$$,

2. 2.

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

3. 3.

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

4. 4.

the triangular graph T(n), with parameters $$(n(n-1)/2,2(n-2),n-2, 4), n \ge 5$$,

5. 5.

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

6. 6.

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

7. 7.

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

8. 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 edge-regular, amply regular, co-edge-regular and triangle-free 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 edge-regular, co-edge-regular 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 co-cliques 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 non-adjacent 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 edge-regular with parameters $$(v,d,\lambda )$$ if R1 holds, amply regular with parameters $$(v,d,\lambda ,\mu )$$ if R1 and R2 hold, co-edge-regular 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(ij) 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 co-clique (independent) set in a graph $$\Gamma$$ is subgraph in which no two vertices are adjacent. A maximum co-clique is a co-clique of largest cardinality. The independence number $$\alpha (\Gamma )$$ of $$\Gamma$$ is the cardinality of a largest co-clique (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. 1.

If C is a co-clique 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. 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. 1.

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

2. 2.

If $$\Gamma$$ and $$\Gamma '$$ are co-edge-regular graphs with parameters $$(v,k,\mu )$$ and $$(v',k',\mu ')$$ where $$v-v' = k - k' = \mu - \mu '$$, then their complete union is co-edge-regular with parameters $$(v + v', k + v', \mu + v').$$

### Definition 2.4

Let $$\Gamma =(V,E)$$ be an edge-regular 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. 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. 2.

If $$\;\Gamma$$ is edge-regular, then $$\Gamma$$ is strongly regular or the line graph of a regular triangle-free graph.

3. 3.

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

4. 4.

If $$\;\Gamma$$ is co-edge-regular, 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 edge-regular 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 edge-regular graphs

### Definition 4.1

Let $$\Gamma =(V,E)$$ be an edge-regular 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 k-subsets of V, $$2\le k \le v-1$$, called blocks, satisfying the following conditions:

1. 1.

$$|B|=k$$ for each $$B\in {\mathcal {B}}$$;

2. 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 edge-regular 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:

\begin{aligned} (v -1-d)\zeta +d \lambda= & {} r (k-1), \end{aligned}
(4.1)
\begin{aligned} vr= & {} bk, \end{aligned}
(4.2)

where $$b=|{\mathcal {B}}|$$.

### Proof

Fix $$i\in V$$ and count in two ways pairs (jB), 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 (iB) , 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 edge-regular 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 $${\overline{A}}=J-I-A$$, 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

\begin{aligned} x_B(i)={\left\{ \begin{array}{ll} 1 \text { if } i\in B\\ 0 \text { if } i\notin B. \end{array}\right. } \end{aligned}

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

\begin{aligned} f_i=\sum _{B \ni i}^{} x_B -\lambda . \end{aligned}

Then, for $$i,j \in V$$,

\begin{aligned} f_i(j)={\left\{ \begin{array}{ll} r-\lambda , \text { if } i=j \\ 0, \text { if } i\ne j \text { and }\{i,j\} \in E\\ \zeta -\lambda , \text { if } i\ne j \text { and } \{i,j\} \notin E. \end{array}\right. } \end{aligned}

Claim 1: At least $$v-1$$ 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

\begin{aligned} \alpha _j (r-\lambda )+(\zeta -\lambda ) \sum _{i\ne j,\; \{i,j\}\notin E } \alpha _i =0. \end{aligned}
(4.3)

Put $${\overline{a}}=(\alpha _1,\alpha _2, \ldots , \alpha _v)^t$$. Then, using (4.3), we get

\begin{aligned} (r-\lambda ) \left( \begin{array}{c} \alpha _1 \\ \alpha _2 \\ \vdots \\ \alpha _v \end{array}\right) = (\lambda -\zeta ) \left( \begin{array}{c} \displaystyle \sum _{\{i,1\}\notin E } \alpha _i \\ \displaystyle \sum _{\{i,2\}\notin E }^{} \alpha _i \\ \vdots \\ \displaystyle \sum _{\{i,v\}\notin E } \alpha _i \end{array}\right) = (\lambda -\zeta ) (J-I-A) \left( \begin{array}{c} \alpha _1 \\ \alpha _2 \\ \vdots \\ \alpha _v \end{array}\right) , \end{aligned}

i.e., $$(J-I-A){\overline{a}} = \left( \dfrac{r-\lambda }{\lambda -\zeta }\right) {\overline{a}}$$, 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

\begin{aligned} {\overline{0}}=\sum _{i=1}^{v} \alpha _i f_i =(\alpha _1,\alpha _2, \ldots , \alpha _v) \left( \begin{array}{c} f_1 \\ f_2 \\ \vdots \\ f_v \end{array}\right) =c (\beta _1,\beta _2,\ldots ,\beta _v) \left( \begin{array}{c} f_1 \\ f_2 \\ \vdots \\ f_v \end{array}\right) . \end{aligned}

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 $$\dim \ge v-1$$.

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,

\begin{aligned} \sum _{B\in {\mathcal {B}}}^{} x_B=\sum _{i=1}^{v} \alpha _i \left( \sum _{B\ni i}^{} x_B -\lambda \right) . \end{aligned}
(4.4)

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

\begin{aligned} r=\sum _{\{i,j\}\in E}^{} \alpha _i -\lambda \sum _{i=1}^{v} \alpha _i=\sum _{\{i,j\}\in E}^{} \alpha _i. \end{aligned}
(4.5)

Summing up (4.5) for $$j=1,2,\ldots ,v$$, we get

\begin{aligned} vr =\sum _{j=1}^{v} \left( \sum _{\{i,j\}\in E}^{} \alpha _i\right) =c \sum _{i=1}^{v} \alpha _i=0, \end{aligned}

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 $$b\ge v-1$$. Now, if $$b=v-1$$, then $$vr=bk$$ gives $$vr=(v-1)k$$, 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 edge-regular 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. 1.

$$r\ge 2 \lambda -\zeta$$;

2. 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 $$(y,z)\notin E$$,

1. (a)

any block containing x must contain y or z, and

2. (b)

any block containing y and z must contain x.

### Remark 4.5

1. 1.

Let $$\Gamma$$ be an edge-regular 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}{(v-1-d)\zeta +d\lambda +r} \end{aligned}

and is an increasing function of r.

2. 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 $${\overline{A}}=J-I-A$$, the adjacency matrix of the complement of $$\Gamma$$.

3. 3.

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

4. 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 triangle-free graph with valency k and $$\Gamma =(V,E)=L(G)$$ be the line graph of G. Then, $$\Gamma$$ is an edge-regular $$(v,d,\lambda )=(|E|k/2,2(k-1), k-2)$$ 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, $$r=2, d=2(k-1), \zeta =0, |{\mathcal {B}}|=|V(G)|, \lambda =1$$ and $$k=3$$.

### Proof

As G is triangle-free, 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 $$k-2$$ edges other than i and j incident on x. Hence, $$\lambda =k-2$$. 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 triangle-free graph with valency $$\ell \ge 3$$ and $$\Gamma =L(G)$$ be the line graph of G. Then, $$\Gamma$$ is an edge-regular $$(v,d,\lambda )=(|V(G)|\ell /2,2(\ell -1), \ell -2)$$ graph. Let $$D=(E(G),{\mathcal {B}})$$ be a $$(v,k,\lambda ,\zeta )$$-design over $$\Gamma$$, where $${\mathcal {B}}$$ is a set of 3-subsets 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 triangle-free 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 edge-regular $$(2n, 2(n-1), 2(n-2))$$-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 $$2^{n-1}$$ maximum cliques in $$\Gamma$$.

Then, $$(V(\Gamma ),{\mathcal {B}})$$ is a $$(2n,n, 2(n-2),0)$$-design, with parameters $$b=8(n-2)$$ and $$r=4(n-2)$$. If $${\mathcal {B}}={\mathcal {B}}'$$, then $$8(n-2)=2^n$$, and hence, $$n=3,4$$.

Suppose $$(V(\Gamma ),{\mathcal {B}})$$ exists for $$n\ge 5$$. Consider $${\mathcal {B}}''={\mathcal {B}}'- {\mathcal {B}}$$, then $$|{\mathcal {B}}''|=2^n-8(n-2)$$ and each vertex of $$\Gamma$$ is contained in $$2^{n-1}-4(n-2)$$ maximum cliques in $${\mathcal {B}}''$$. Count the pairs (xB), where $$x\in V(\Gamma )$$ and $$B\in {\mathcal {B}}''$$, in two different ways, to see that

\begin{aligned} (2^n-8(n-2))n=2^n(2^{(n-1)}-4(n-2)). \end{aligned}
(4.6)

This implies $$n\le 8$$ as $$2^n$$ divides $$8(n-2)n$$. 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 edge-regular 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 $$d=2(k-1)$$.

### 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 $$d=2(k-1)$$, and using equation (4.2), we get $$b= 8 \lambda -\dfrac{2 \lambda (\lambda +4)}{k}$$. Hence, $$v=4 (k-1)-\lambda$$. $$\square$$

## 5 Examples of designs over strongly regular graphs considered as edge-regular 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 $$v-k=\dfrac{(2 k-d-2) (\lambda -\zeta )}{\zeta }$$. 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 edge-regular graphs

In this section, we give possible parameters of designs associated with edge-regular 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 edge-regular graphs.

1. (a)

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

2. (b)

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

3. (c)

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

4. (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 co-edge-regular 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 co-edge-regular 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 k-subsets of V, $$2\le k \le v-1$$, called blocks, satisfying the following conditions:

1. 1.

$$|B|=k$$ for each $$B\in {\mathcal {B}}$$;

2. 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 co-edge-regular 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:

\begin{aligned} d\xi +(v-d-1) \mu= & {} r(k-1), \end{aligned}
(7.1)
\begin{aligned} vr= & {} bk, \end{aligned}
(7.2)

where $$b=|{\mathcal {B}}|$$.

### Proof

We obtain (7.1) by fixing $$i\in V$$, and counting in two ways pairs (jB), where $$j\in V, j\ne i,$$ and $$B\in {\mathcal {B}}$$ with $$\{i,j\} \subseteq B$$. By counting in two ways pairs (jB), 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 edge-regular case.

### Theorem 7.3

Let $$\Gamma =(V,E)$$ be a co-edge-regular 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 $${\overline{A}}=J-I-A$$, 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 co-edge-regular graph $$\Gamma =(V,E)$$ with parameters $$(v,d,\mu )$$, where $$\mu \ge 1$$ and r be the replication number of D. Then,

1. 1.

$$r\ge 2 \mu -\xi$$;

2. 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 $$(y,z)\in E$$,

1. (a)

any block containing x must contain y or z, and

2. (b)

any block containing y and z must contain x.

We can make the following observations concerning a $$(v,k,\mu ,\xi )$$-design over a co-edge-regular graph $$\Gamma$$ with parameters $$(v,d,\mu )$$.

### Remark 7.5

1. 1.

Let $$\Gamma$$ be a co-edge-regular 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 +(v-d-1)\mu } \end{aligned}

and is an increasing function of r.

2. 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 $${\overline{A}}=J-I-A$$, 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 co-edge-regular graphs with least eigenvalue $$-2$$.

3. 3.

$${\mathcal {B}}$$ is the set of co-cliques of $$\Gamma$$ if and only if $$\xi =0$$.

4. 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 co-clique 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 co-cliques of $$\Gamma$$, be a $$(v,k,\mu ,\xi )$$-design over a co-edge-regular 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 co-cliques of $$\Gamma$$. Suppose $$r=\mu$$, then $$\xi =0$$ and the relation $$(v-d-1)\mu =\mu (k-1)$$ gives $$k=v-d$$. Let S be a set of maximum co-cliques 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 co-clique, namely S itself, which implies that a vertex in $$V - S$$ does not belong to any maximum co-clique, 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 co-edge-regular graphs

In this section, we try to find examples of designs over co-edge-regular graph with smallest eigenvalue $$-2$$ and $$r= 2 \mu -\xi$$. If $$\xi =0$$, then we explore the possibility for blocks as co-cliques of $$\Gamma$$. We use equation (7.1) to see that $$v-k=\frac{(d+k-1) (\mu -\xi )}{\mu }$$. 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 co-edge-regular graph with parameters $$(mn,m+n-2,2)$$.

For $$\xi =0$$, observe that $$b-12=-\dfrac{4 (m-3) (n-3)}{(m-1) (n-1)+2}$$ and $$k=m+\dfrac{1}{2} (m-1) (n-3)$$. As blocks are co-cliques of $$\Gamma , k\le m$$ and $$\Gamma$$ has $$n(n-1)\cdots (n-m+1)$$ co-cliques of size m. So $$n\le 3$$. We list the possible values of other parameters in Table 3.

For $$\xi =1$$, observe that $$b=\dfrac{9 m n}{2 n m-m-n+3}, r=3$$ and $$k=\dfrac{1}{3} (2 n m-m-n+3)$$. As $$\dfrac{ m n}{2 n m-m-n+3}<1$$, $$b\le 8$$. We write $$m=\dfrac{b (n-3)}{(2 n -1)b-9 n}$$ and observe that $$m=1$$. We list the possible values of other parameters in Table 3.

## 9 Possible parameters of designs associated with co-edge-regular graphs

In this section, we give possible parameters of designs associated with co-edge-regular graph $$\Gamma$$ with $$r= 2 \mu -\xi$$ and $${\overline{\Gamma }}$$ having smallest eigenvalue $$-2$$. As before, we consider $$0\le \xi < \mu$$. We refer [2] for detail of graphs listed in Table 6.

## 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

\begin{aligned} k_2\mu =d(d-1-\lambda ). \end{aligned}
(10.1)

### 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 k-subsets of V$$2\le k \le v-1$$, called blocks, satisfying the following conditions:

1. 1.

if $$i,j\in V$$, $$i\ne j$$, and $$\{i,j\}\in E$$, then there are exactly $$\lambda$$ blocks containing $$\{i,j\},$$

2. 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. 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

\begin{aligned} d\lambda +k_2\mu + (v-1-d-k_2) \xi= & {} r(k-1), \end{aligned}
(10.2)
\begin{aligned} vr= & {} bk, \end{aligned}
(10.3)

where $$b=|{\mathcal {B}}|$$.

### Proof

We obtain (10.2) by fixing $$i\in V$$ and counting in two ways, pairs (jB), where $$j\in V$$, $$i\ne j$$, $$B\in {\mathcal {B}}$$ and $$\{i,j\}\subset B$$. By counting in two ways pairs (jB), 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

\begin{aligned} x_B(i)= {\left\{ \begin{array}{ll} 1 \text { if } i \in B\\ 0 \text { if } i \notin B. \end{array}\right. } \end{aligned}

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

\begin{aligned} f_i=\sum _{B \ni i}^{} x_B -\xi . \end{aligned}

Then, for $$i,j \in V$$,

\begin{aligned} f_i(j)={\left\{ \begin{array}{ll} r-\xi \text { if } i=j \\ \lambda -\xi \text { if } i\ne j \text { and }\{i,j\} \in E\\ \mu -\xi \text { if } i\ne j \text { and } d(i,j) = 2\\ 0 \text { if } i\ne j \text { and } d(i,j) > 2. \end{array}\right. } \end{aligned}

We claim that at least $$v-1$$ 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

\begin{aligned} \alpha _j (r-\xi )+(\lambda -\xi ) \sum _{i\ne j,\; \{i,j\}\subset E } \alpha _i+ (\mu -\xi ) \sum _{i\ne j,\; d(i,j)=2 } \alpha _i+ 0\sum _{i\ne j, \; d(i,j) \ge 2 } \alpha _i =0.\nonumber \\ \end{aligned}
(10.4)

Let $${\overline{a}}=(\alpha _1,\alpha _2, \ldots , \alpha _v)$$. Then, we can rewrite (10.4) as

\begin{aligned}&(r-\xi ) {\overline{a}} +(\lambda -\xi ) A {\overline{a}}+ (\mu -\xi ) \mu (A^2-\lambda A-d I) {\overline{a}}=0, \\&\quad (r-\xi -d\mu (\mu -\xi )){\overline{a}} +(\lambda -\xi - (\mu -\xi ) \mu \lambda ) A {\overline{a}}+ (\mu -\xi ) \mu A^2 {\overline{a}}=0, \end{aligned}

which gives

\begin{aligned} ((\lambda -\xi - (\mu -\xi ) \mu \lambda ) A + (\mu -\xi ) \mu A^2)) {\overline{a}}= (d\mu (\mu -\xi )-(r-\xi )) {\overline{a}}. \end{aligned}

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 $$\ge v-1$$.

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

\begin{aligned} \sum _{B \in {\mathcal {B}}}^{} x_B = \sum _{i=1}^{v} \alpha _i \left( \sum _{B \ni i}^{} x_B- \xi \right) . \end{aligned}
(10.5)

Then, equating the constant term on both sides of (10.5) of the point $$j\in V$$, we obtain

\begin{aligned} r=\alpha _j (r-\xi ) +(\lambda -\xi ) \sum _{\{i,j\}\subset E}^{} \alpha _i+ (\mu -\xi ) \sum _{d(i,j)=2}^{} \alpha _i. \end{aligned}
(10.6)

Summing (10.6) for $$j=1,2, \ldots ,v$$, gives

\begin{aligned} vr=(r-\xi ) \sum _{j=1}^{v} \alpha _j + (\lambda -\xi ) d \sum _{j=1}^{v} \alpha _j+ (\mu -\xi ) k_2 \sum _{j=1}^{v} \alpha _j. \end{aligned}
(10.7)

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 $$b\ge v-1$$. Now, if $$b=v-1$$, then (10.3) implies that $$v-1$$ divides r, so $$r=v-1$$ and $$k=v-1$$, 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. 1.

$$r\ge 2 \lambda -\mu$$;

2. 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 $$d(y,z)=2$$,

1. (a)

any block containing x must contain y or z, and

2. (b)

any block containing y and z must contain x.

### Proof

Let xyz 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. 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 + (v-1-d-k_2) \xi +r} \end{aligned}

and is an increasing function of r.

2. 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 edge-regular graphs, amply regular graphs and co-edge-regular 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 non-strongly regular graphs. In the case of amply regular graphs, construction of designs is challenging. The combination of cliques and co-cliques may obtain examples of these types of designs. To find cliques or co-cliques of a graph and also to perform other graph operations, we have used computer algebra system SageMath [10].