1 Introduction

In this paper, we only consider finite graphs. Let Γ=(X,R) be a connected graph with vertex set X and edge set R consisting of 2-element subsets of X. When {x,y}∈R, i.e., x and y are adjacent, we write x∼y. For x,y∈X, ∂ Γ (x,y)=∂(x,y) denotes the distance between x and y, i.e., the length of a shortest path between x and y in Γ. The diameter d(Γ) is the maximal distance between two vertices. A nonempty subset C of X is said to be a clique if every distinct vertices in C are adjacent.

A subset C of X is often called a code in Γ=(X,R), and Γ i (C)={x∈X∣∂(x,C)=i} is called the ith subconstituent with respect to C, where ∂(x,C)=min{∂(x,y)∣y∈C}. We write Γ(C) for Γ 1(C). The number t=t(C)=max{i∣Γ i (C)≠∅} is called the covering radius of C. If C and C′ are subsets of X, ∂(C,C′)=min{∂(x,y)∣x∈C,y∈C′}. When C={x}, we write Γ i (x) for Γ i ({x}), and set Γ(x) for Γ 1(x). The number k(x)=|Γ(x)| is called the valency of x. For x,y∈X with ∂(x,y)=i with i=0,1,…,d(Γ), let

$$\begin{aligned} B_i(x,y) &= \varGamma_{i+1}(x)\cap\varGamma(y), \qquad A_i(x,y) = \varGamma _{i}(x)\cap \varGamma(y), \\ C_i(x,y) &= \varGamma_{i-1}(x)\cap\varGamma(y), \end{aligned}$$

and b i (x,y)=|B i (x,y)|, a i (x,y)=|A i (x,y)| and c i (x,y)=|C i (x,y)|.

A connected graph Γ=(X,R) of diameter D=d(Γ) is said to be distance-regular if for each i∈{0,1,2,…,D}, the numbers c i =c i (x,y), a i =a i (x,y) and b i =b i (x,y) depend only on i=∂(x,y). In this case, the numbers b i ,a i ,c i with i=0,1,…,D are called the parameters of Γ. For distance-regular graphs we refer the reader [4]. We mainly follow the notation and the terminologies in the monograph.

Let Γ=(X∪Y,R) be a connected bipartite graph with the bipartition X∪Y, i.e., there is no edge within X and Y. Let d X=d X(Γ)=max{∂(x,y)∣x∈X,y∈X∪Y}. Then Γ is said to be distance-semiregular on X, if for each i∈{0,1,2,…,d X}, the numbers \(c^{X}_{i} = c_{i}(x,y)\), and \(b^{X}_{i} = b_{i}(x,y)\) depend only on i=∂(x,y) whenever x∈X and y∈X∪Y. In this case the numbers \(b^{X}_{i}, c^{X}_{i}\) with i=0,1,…,d X are called the parameters of Γ. Note that each vertex y∈Y is of valency \(b^{Y}_{0} = b^{X}_{1}+c^{X}_{1}\) and distance-semiregular graphs are biregular, i.e., the valency of a vertex depends only on the part the vertex belongs to. If Γ=(X∪Y,R) is distance-semiregular on both X and Y, Γ is called distance-biregular. For more information on distance-biregular graphs and distance-semiregular graphs, see [9, 11, 14].

Let Γ=(X,R) be a connected graph, and C a nonempty subset of X with covering radius t=t(C). Then C is said to be a completely regular code if γ i =γ i (C)=|Γ i−1(C)∩Γ(x)|, α i =α i (C)=|Γ i (C)∩Γ(x)|, β i =β i (C)=|Γ i+1(C)∩Γ(x)| do not depend on x∈Γ i (C) for i∈{0,1,…,t}. In this case the numbers γ i ,α i ,β i with i=0,1,…,t are called the parameters of C. We also write γ i (C), α i (C) or β i (C) exists when the corresponding number does not depend on the choice of x∈Γ i (C). For completely regular codes of distance-regular graphs, see [4, Sect. 11.1] and [13]. For a special type of completely regular codes in a regular graph, see [7].

Let Γ=(X,R) be a connected graph. In [11], C.D. Godsil and J. Shawe-Taylor showed that {x} is completely regular for each x∈X, if and only if Γ is either distance-regular or distance-biregular. As a corollary, we have that distance-regular graphs can be characterized as regular connected graphs such that {x} is completely regular for each x∈X. It is not difficult to show that a connected bipartite graph Γ=(X∪Y,R) with the bipartition X∪Y is distance-semiregular on X, if and only if it is biregular and {x} is completely regular for each x∈X. Recently it was shown in [6] that each edge of Γ is completely regular with the same parameters, if and only if Γ is either bipartite or almost bipartite distance-regular, i.e., distance-regular graphs of diameter D with a 0=a 1=⋯=a D−1=0. These results can be viewed as characterizations of distance-regularity by complete regularity of its substructures, i.e., cliques.

On the other hand, many distance-regular graphs contain many completely regular codes of various sizes. These completely regular codes correspond to substructures of the geometry associated with them. If Γ is a distance-regular graph of diameter D isomorphic to one of the Johnson graphs, the Hamming graphs, the Grassmann graphs, the dual polar graphs, the bilinear forms graphs, then for each i∈{0,1,…,D} and vertices x,y at distance i, there is a geodetically closed completely regular code of diameter i containing x and y. In each case, there is a set \(\mathcal{C}\) of completely regular maximal cliques such that the incidence graph on \(X\cup\mathcal{C}\) associated with it is distance-semiregular.

Definition 1

Let Γ=(X,R) be a connected graph, and let \(\mathcal{C}\) be a collection of cliques of Γ. Then Γ is said to be a completely regular clique graph with parameters (s,c) with respect to \(\mathcal{C}\), if the following are satisfied.

  1. (i)

    Each member \(C\in\mathcal{C}\) is a completely regular code of size s+1≥2.

  2. (ii)

    Each edge is contained in exactly c members of \(\mathcal{C} \) and c≥1.

When \(\mathcal{C}\) consists of Delsarte cliques, it is called a Delsarte clique graph with parameters (s,c) in [2, 3], and Delsarte clique graphs with parameters (s,1) are called geometric in [1]. Many examples are listed in [2].

Our first result in this paper is concerning the parameters of \(C\in \mathcal{C}\).

Theorem 1

Let Γ be a completely regular clique graph with parameters (s,c) with respect to \(\mathcal{C}\). Then the parameters of a completely regular code \(C\in\mathcal{C}\) do not depend on C.

We prove Theorem 1 in Sect. 2 using modules of Terwilliger algebra \(\mathcal{T}(C)\) with respect to \(C\in\mathcal {C}\), and applying the fact that C is completely regular if and only if the primary \(\mathcal{T} (C)\)-module is thin. For Terwilliger algebras \(\mathcal{T}(C)\) and their modules, see [15].

A connected graph is called edge distance-regular if every edge is a completely regular code with the same parameters. See [5, 10]. Because of Theorem 1, the condition on parameters is not necessary. Combining with the result in [6] mentioned above, we have the following.

Corollary 2

Let Γ=(X,R) be a connected graph of diameter D. Then the following are equivalent.

  1. (i)

    For every {x,y}∈R, {x,y} is a completely regular code.

  2. (ii)

    Γ is a distance-regular graph with a 1=⋯=a D−1=0.

Next result is a characterization of distance-regular completely regular clique graphs. The proof will be given in Sect. 3. It can be viewed as a characterization of the collinearity graphs of distance-regular geometries in [8, 12].

Theorem 3

Let Γ=(X,R) be a distance-regular graph. Then Γ is a completely regular clique graph if and only if Γ is the bipartite half of a distance-semiregular graph \(\tilde{\varGamma} = (X\cup Y, \tilde{R})\) on X. Here the bipartite half of \(\tilde{\varGamma} = (X\cup Y, \tilde{R})\) on X is the graph with vertex set X such that two vertices are adjacent whenever they are at distance 2 in \(\tilde {\varGamma}\).

2 Parameter set of completely regular clique codes

The main objective of this section is to prove Theorem 1.

Lemma 4

Let Γ=(X,R) be a connected graph of diameter D>1. Let \(\mathcal{C}\) be a collection of cliques of Γ. Then the following hold.

  1. (i)

    Let \(C, C'\in\mathcal{C}\) with C∩C′≠∅. Suppose α 0(C),β 0(C),α 0(C′) and β 0(C′) exist. Then the valencies of the vertices in C∪C′ are the same.

  2. (ii)

    If every edge is contained in at least one \(C\in \mathcal{C} \), and both α 0(C) and β 0(C) exist for all \(C\in \mathcal{C}\), then Γ is regular with valency k=α 0(C)+β 0(C) for any \(C\in\mathcal{C}\).

Proof

(i) For all x,y∈C, k(x)=β 0(C)+α 0(C)=k(y), and for all x′,y′∈C′, k(x′)=β 0(C′)+α 0(C′)=k(y′). Since C∩C′≠∅, the valency k(x) is constant on C∪C′.

(ii) Since Γ is connected, the assertion follows from (i). □

Let Γ=(X,R) be a connected graph of diameter D and C a nonempty subset of X with covering radius t(C). Let V=R X denote the real vector space consisting of column vectors whose entries are indexed by X. For u,v∈V, 〈u,v〉=u T v and \(\|\boldsymbol{u}\| = \sqrt{\langle \boldsymbol{u}, \boldsymbol{u}\rangle}\). Let \(A\in \operatorname{Mat}_{X}(\boldsymbol{R}) \) be the adjacency matrix of Γ. Let θ 0>θ 1>⋯>θ r be all the distinct eigenvalues of A, and E 0,E 1,…,E r ∈R[A] the corresponding primitive idempotents, where R[A] is the polynomial algebra in A over the real number field. Thus,

$$\begin{aligned} &E_0+E_1+\cdots+E_r = I, \qquad E_iE_j = \delta_{i,j}E_i, \quad \mbox{and}\\ & AE_i = \theta_iE_i \quad \mbox{for } i, j\in \{0, 1, \ldots, r\}. \end{aligned}$$

For each i∈{0,1,…,t(C)}, let \(E^{*}_{i}(C)\) denote the projection onto the subspace of V spanned by unit vectors corresponding to vertices in Γ i (C). We let \(\mathcal{T}(C)\) denote the subalgebra of \(\operatorname{Mat}_{X}(\boldsymbol{R})\) generated by A and \(E^{*}_{0}(C), E^{*}_{1}(C)\), \(\ldots, E^{*}_{t(C)}(C)\). Let 1∈V be the all one vector. A \(\mathcal{T} (C)\)-module W, i.e., a vector subspace of V invariant under the action of \(\mathcal{T}(C)\), is said to be thin if \(\dim E^{*}_{i}(C)W \leq1\) for all i∈{0,1,…,t(C)}. \(\mathcal{T}(C)\boldsymbol {1}\) is called the primary module of \(\mathcal{T}(C)\). Note that \(\boldsymbol{w}_{i} = E^{*}_{i}(C)\boldsymbol{1}\) is the characteristic vector of Γ i (C) for i∈{0,1,…,t(C)} and \(W_{C} = \operatorname{Span}(\boldsymbol{w}_{0}, \boldsymbol{w}_{1}, \ldots, \boldsymbol{w}_{t(C)}) \subset\mathcal{T} (C)\boldsymbol{1}\). It is easy to see that if C is a completely regular code with parameters γ i ,α i ,β i (i=0,1,…,t(C)), then

$$A\boldsymbol{w}_i = \beta_{i-1}\boldsymbol{w}_{i-1} + \alpha _i\boldsymbol{w}_i + \gamma_{i+1} \boldsymbol{w}_{i+1}\quad \mbox{for } i = 0, 1, \ldots, t(C). $$

Here w −1=w t(C)+1=0, and β −1 and γ t(C)+1 are indeterminate. Hence in this case \(W_{C} = \mathcal {T}(C)\boldsymbol{1}\) and W C is a thin irreducible \(\mathcal{T}(C)\)-module. See [15, Proposition 7.2].

The ideas and techniques of proofs of the following results are taken from the lecture note by P. Terwilliger [16].

Proposition 5

Let Γ=(X,R) be a connected graph of diameter D>1. Suppose C,C′ are completely regular codes that are cliques of Γ with |C∩C′|=e>0. Let

$$\begin{aligned} W =& W_C = \operatorname{Span}(\boldsymbol{w}_0, \boldsymbol {w}_1, \ldots, \boldsymbol{w}_t), \quad \textit{where } \boldsymbol{w}_i = E^*_i(C)\boldsymbol{1}, \\ A\boldsymbol{w}_i = & \beta_{i-1}\boldsymbol{w}_{i-1} + \alpha _i\boldsymbol{w}_i + \gamma _{i+1} \boldsymbol{w} _{i+1}, \\ W' = &W_{C'} = \operatorname{Span}\bigl( \boldsymbol{w}'_0, \boldsymbol {w}'_1, \ldots, \boldsymbol{w}'_{t'}\bigr), \quad \textit{where } \boldsymbol{w}'_i = E^*_i \bigl(C'\bigr)\boldsymbol{1}, \\ A\boldsymbol{w}'_i = & \beta'_{i-1} \boldsymbol{w}'_{i-1} + \alpha '_i \boldsymbol{w}'_i + \gamma '_{i+1} \boldsymbol{w}'_{i+1}, \end{aligned}$$

with t=t(C) and t′=t(C′). Then the following hold.

  1. (i)

    There are polynomials p(λ),p′(λ)∈R[λ] such that

    $$\operatorname{proj}_{W'}\boldsymbol{w}_0 = p(A) \frac{\|\boldsymbol {w}_0\|}{\|\boldsymbol{w}'_0\|}\boldsymbol{w}'_0, \qquad \operatorname{proj} _{W}\boldsymbol{w} '_0 = p'(A) \frac{\|\boldsymbol{w}'_0\|}{\|\boldsymbol{w}_0\| }\boldsymbol{w}_0, $$

    where with k=α 0+β 0,

    $$\begin{aligned} p(\lambda) = & \frac{\|\boldsymbol{w}_0\|^2-e}{(k-\|\boldsymbol {w}'_0\|^2+1)\|\boldsymbol{w}_0\|\| \boldsymbol{w} '_0\|}\lambda+ \frac{ke - \|\boldsymbol{w}_0\|^2(\|\boldsymbol {w}'_0\|^2-1)}{(k-\|\boldsymbol{w} '_0\| ^2+1)\|\boldsymbol{w}_0\|\|\boldsymbol{w}'_0\|}, \\ p'(\lambda) = & \frac{\|\boldsymbol{w}'_0\|^2-e}{(k-\|\boldsymbol {w}_0\|^2+1)\|\boldsymbol{w}_0\| \|\boldsymbol{w} '_0\|}\lambda+ \frac{ke - \|\boldsymbol{w}'_0\|^2(\|\boldsymbol {w}_0\|^2-1)}{(k-\|\boldsymbol{w}_0\| ^2+1)\|\boldsymbol{w}_0\|\|\boldsymbol{w}'_0\|}. \end{aligned}$$
  2. (ii)

    p′(θ i )m W (θ i )=p(θ i )m W′(θ i ) for all i, where

    $$m_W(\theta_i) = \frac{\|E_i\boldsymbol{w}_0\|^2}{\|\boldsymbol {w}_0\|^2}\quad \textit{and}\quad m_{W'}(\theta_i) = \frac{\|E_i\boldsymbol{w}'_0\|^2}{\|\boldsymbol {w}'_0\|^2}. $$
  3. (iii)

    If \(\|\boldsymbol{w}_{0}\| = \|\boldsymbol{w}'_{0}\| \), i.e., |C|=|C′|, then m W (θ i )=m W′(θ i ) for all i.

Proof

Since C∩C′≠∅, the valencies of the vertices in C∪C′ are the same by Lemma 4. Let \(k = \alpha _{0} + \beta_{0} = \alpha'_{0} + \beta'_{0}\) be the valency of vertices in C∪C′.

(i) Since A w 0=α 0 w 0+γ 1 w 1 with γ 1≠0 and α 0=|C|−1=∥w 0∥2−1,

$$\boldsymbol{w}_1 = \frac{1}{\gamma_1}\bigl(A-\bigl(\| \boldsymbol{w}_0\| ^2-1\bigr)I\bigr)\boldsymbol{w}_0. $$

Since C′⊂C∪Γ(C), \(\langle\boldsymbol{w}'_{0}, \boldsymbol{w}_{i}\rangle= 0\) for i=2,3,…,t. Let \(\operatorname{proj}_{W}\boldsymbol {w}'_{0} = \xi_{0}\boldsymbol{w}_{0} + \xi _{1}\boldsymbol{w} _{1}\). By counting the edges between C and Γ(C) in two ways, we have

$$\|\boldsymbol{w}_1\|^2 = \langle\boldsymbol{w}_1, \boldsymbol {w}_1\rangle= \bigl|\varGamma(C)\bigr| = \frac {\beta _0|C|}{\gamma_1} = \frac{(k-\|\boldsymbol{w}_0\|^2+1)\|\boldsymbol {w}_0\|^2}{\gamma_1}, $$

\(e = \langle\boldsymbol{w}'_{0}, \boldsymbol{w}_{0}\rangle= \langle \operatorname{proj}_{W}\boldsymbol{w}'_{0}, \boldsymbol{w} _{0}\rangle= \xi_{0}\|\boldsymbol{w}_{0}\|^{2}\), and

$$\bigl\| \boldsymbol{w}'_0\bigr\| ^2-e = \bigl\langle \boldsymbol{w}'_0, \boldsymbol {w}_1\bigr\rangle = \bigl\langle \operatorname{proj} _{W}\boldsymbol{w} '_0, \boldsymbol{w}_1\bigr\rangle = \xi_1\|\boldsymbol{w}_1\|^2 = \xi _1\frac{(k-\|\boldsymbol{w}_0\| ^2+1)\|\boldsymbol{w} _0\|^2}{\gamma_1}. $$

Therefore,

$$\begin{aligned} \begin{aligned} &\operatorname{proj}_{W}\boldsymbol{w}'_0\\ &\quad = \xi_0\boldsymbol{w}_0 + \xi_1 \boldsymbol{w}_1 \\ &\quad = \frac{e}{\|\boldsymbol{w}_0\|^2}\boldsymbol{w}_0 + \frac {\gamma_1(\|\boldsymbol{w}'_0\| ^2-e)}{(k-\| \boldsymbol{w}_0\|^2+1)\|\boldsymbol{w}_0\|^2}\cdot \frac{1}{\gamma _1}\bigl(A-\bigl(\|\boldsymbol{w}_0\| ^2-1 \bigr)I\bigr)\boldsymbol{w} _0 \\ &\quad = \frac{\|\boldsymbol{w}'_0\|}{\|\boldsymbol{w}_0\|} \biggl(\frac {\|\boldsymbol{w}'_0\| ^2-e}{(k-\|\boldsymbol{w} _0\|^2+1)\|\boldsymbol{w}_0\|\|\boldsymbol{w}'_0\|}A + \frac{ke - \| \boldsymbol{w}'_0\|^2(\|\boldsymbol{w}_0\| ^2-1)}{(k-\|\boldsymbol{w}_0\|^2+1)\|\boldsymbol{w}_0\|\|\boldsymbol {w}'_0\|}I \biggr)\boldsymbol{w}_0 \\ &\quad = p'(A)\frac{\|\boldsymbol{w}'_0\|}{\|\boldsymbol{w}_0\| }\boldsymbol{w}_0. \end{aligned} \end{aligned}$$

By symmetry we obtain the formula of \(\operatorname {proj}_{W'}\boldsymbol{w}_{0}\) as well.

(ii) Since E i w 0∈W and A is a real symmetric matrix,

$$\begin{aligned} \frac{\langle E_i\boldsymbol{w}_0, E_i\boldsymbol{w}'_0\rangle}{\| \boldsymbol{w}_0\|\|\boldsymbol{w}'_0\|} = & \frac{\langle E_i\boldsymbol{w}_0, \boldsymbol{w}'_0\rangle}{\| \boldsymbol{w}_0\|\|\boldsymbol{w}'_0\|} = \frac {\langle E_i\boldsymbol{w}_0, \operatorname{proj}_{W}\boldsymbol {w}'_0\rangle}{\|\boldsymbol{w}_0\|\|\boldsymbol{w}'_0\|} = \frac {\|\boldsymbol{w}'_0\|}{\|\boldsymbol{w}_0\|}\frac{\langle E_i\boldsymbol{w}_0, p'(A)\boldsymbol{w}_0\rangle }{\|\boldsymbol{w} _0\|\|\boldsymbol{w}'_0\|} \\ = & \frac{\langle p'(A)E_i\boldsymbol{w}_0, \boldsymbol {w}_0\rangle}{\|\boldsymbol{w}_0\|^2} = p'(\theta_i) \frac{\langle E_i\boldsymbol{w}_0, E_i\boldsymbol {w}_0\rangle}{\|\boldsymbol{w}_0\|^2} = p'(\theta_i)\frac{\|E_i\boldsymbol{w}_0\|^2}{\|\boldsymbol{w}_0\| ^2} \\ = & p'(\theta_i)m_W( \theta_i). \end{aligned}$$

By symmetry we have (ii).

(iii) Suppose \(\|\boldsymbol{w}_{0}\| = \|\boldsymbol{w}'_{0}\|\). Then p(λ)=p′(λ) by (i). Therefore m W (θ i )=m W′(θ i ) for all i except possibly one i for which p(θ i )=0. Since

$$1 = \frac{\langle\boldsymbol{w}_0,\boldsymbol{w}_0\rangle}{\| \boldsymbol{w}_0\|^2} = \sum_{i=0}^r \frac {\|E_i\boldsymbol{w}_0\|^2}{\|\boldsymbol{w}_0\|^2} = \sum_{i=0}^r m_W(\theta_i), $$

m W (θ i )=m W′(θ i ) for all i without an exception. □

Proposition 6

Let Γ=(X,R) be a connected graph of diameter D>1. Suppose Γ is regular of valency k. Let C,C′ be completely regular codes that are cliques in Γ with C∩C′≠∅. If |C|=|C′|, then the parameters of C and C′ coincide.

Proof

We use the notation in the proof of Proposition 5. By Proposition 5(iii), m W (θ i )=m W′(θ i ) for all i. Since E i E j =δ i,j E i for i,j∈{0,1,…,r}, nonzero vectors in the set {E 0 w 0,E 1 w 0,…,E r w 0} are perpendicular to each other, and hence they form a linearly independent set of vectors. Since \(\boldsymbol{R}[A]\boldsymbol{w}_{0} = \operatorname{Span}\{E_{0}\boldsymbol{w}_{0}, E_{1}\boldsymbol{w} _{0}, \ldots, E_{r}\boldsymbol{w}_{0}\}\), dimR[A]w 0=dim(W)=t(C)+1 is equal to the number of i such that E i w 0≠0, we have t(C)=t(C′). Let t be this number. Let m(θ i )=m W (θ i )=∥E i w 0∥2/∥w 0∥2, and let R t [λ] be the set of all polynomials of degree at most t. Then, for f,g∈R t [λ],

$$\langle f, g \rangle_m = \sum_{i=0}^rf( \theta_i)g(\theta_i)m(\theta_i) $$

defines an inner product on R t [λ]. Let q 0,q 1,…,q t be uniquely determined monic orthogonal polynomials with respect to this inner product. Let p 0,p 1, …,p t ,p t+1 and \(p'_{0}, p'_{1}, \ldots, p'_{t}, p'_{t+1}\) be polynomials defined by the following recursive relations:

$$\begin{aligned} &p_0 = 1, \quad \lambda p_i = \beta_{i-1}p_{i-1} + \alpha_ip_i + \gamma _{i+1}p_{i+1},\quad \mbox{for } i = 0, 1, \ldots,\ t \mbox{ with } p_{-1} = 0,\\ &p'_0 = 1, \quad \lambda p'_i = \beta'_{i-1}p'_{i-1} + \alpha'_ip'_i + \gamma'_{i+1}p'_{i+1},\quad \mbox{for } i = 0, 1, \ldots, \ t \mbox{ with } p'_{-1} = 0. \end{aligned}$$

Here, β i =β i (C), α i =α i (C), γ i =γ i (C), \(\beta'_{i} = \beta_{i}(C')\), \(\alpha'_{i} = \alpha _{i}(C')\), \(\gamma'_{i} = \gamma_{i}(C')\) (i=0,1,…,t), and assume \(\gamma_{t+1} = \gamma'_{t+1} = 1\), \(\beta_{-1} = \beta'_{-1} = 0\). Then we have p i (A)w 0=w i , \(p'_{i}(A)\boldsymbol{w}'_{0} = \boldsymbol{w}'_{i}\) and \(p_{t+1}(A)\boldsymbol{w}_{0} = p'_{t+1}(A)\boldsymbol{w}'_{0} = \boldsymbol{0}\). Moreover, for i,j∈{0,1,…,t+1} with \(\boldsymbol{w}_{t+1} = \boldsymbol {w}'_{t+1} = \boldsymbol{0}\),

$$\begin{aligned} \langle\boldsymbol{w}_i,\boldsymbol{w}_j \rangle = & \bigl\langle p_i(A)\boldsymbol{w}_0,p_j(A) \boldsymbol{w}_0 \bigr\rangle \\ = & \Biggl\langle p_i(A)\sum_{h=0}^r E_h\boldsymbol{w}_0,p_j(A)\sum _{h'=0}^r E_{h'}\boldsymbol{w} _0 \Biggr\rangle \\ = & \sum_{h=0}^r\sum _{h'=0}^r p_i(\theta_h)p_j( \theta _{h'})\langle E_h\boldsymbol{w}_0,E_{h'} \boldsymbol{w}_0 \rangle \\ = & \sum_{h=0}^r p_i( \theta_h)p_j(\theta_{h})m( \theta_h)\| \boldsymbol{w}_0\| ^2 \\ = & \langle p_i,p_j \rangle_m \| \boldsymbol{w}_0\|^2. \end{aligned}$$

Therefore we have for i,j∈{0,1,…,t},

$$\langle p_i,p_j \rangle_m = \delta_{i,j} = \bigl\langle p'_i,p'_j \bigr\rangle _m, $$

and \(\langle p_{t+1},p_{t+1}\rangle_{m} = \langle p'_{t+1},p'_{t+1}\rangle _{m} = 0\). Considering the leading coefficients of p i and p j , we have

$$\gamma_1\gamma_2\cdots\gamma_ip_i = q_i = \gamma'_1\gamma '_2 \cdots\gamma '_ip'_i\quad \mbox{for all } i\in\{0, 1, \ldots, t\}. $$

In particular, q 0,q 2,…q t with q t+1, monic characteristic polynomial of A on W, satisfy

$$\lambda q_i = \beta_{i-1}\gamma_iq_{i-1} + \alpha_iq_i + q_{i+1} = \beta '_{i-1}\gamma'_iq_{i-1} + \alpha'_iq_i + q_{i+1} \quad \mbox{for } i = 0, 1, 2, \ldots, t. $$

Therefore we have

$$\beta_{i-1}\gamma_{i} = \beta'_{i-1} \gamma'_{i} \quad \mbox{and}\quad \alpha_{i} = \alpha'_{i} \quad \mbox{for } i = 0, 1, 2, \ldots, t. $$

Since Γ is regular of valency k, \(\alpha_{0} = \alpha'_{0}\) implies \(\beta_{0} = k - \alpha_{0} = k-\alpha'_{0} = \beta'_{0}\). Hence we have \(\gamma_{1} = \gamma'_{1}\) by above. Since \(\alpha_{i} = \alpha'_{i}\) for all i, by induction, we can conclude that all parameters are equal. □

Proof of Theorem 1

Let Γ be a completely regular clique graph with parameters (s,c) with respect to \(\mathcal{C}\). Since every edge is contained in a member of \(\mathcal{C}\), Γ is regular by Lemma 4. Now the fact that the parameters of completely regular codes \(C\in \mathcal{C}\) do not depend on C follows from Proposition 6. □

Example 1

The prism graph below has two types of completely regular cliques, i.e., triangles {a,b,c} and {d,e,f}, and edges {a,f},{c,d} and {b,e}. Parameters are different. If C={a,b,c} and C′={a,f}, then \(p(\lambda) = \frac{1}{6}\lambda\) and \(p'(\lambda) = \frac{1}{6}(\lambda-1)\).

figure a

Example 2

The following graph is 3-regular with 8 vertices. Both edges {a,b} and {c,d} are completely regular but parameters are different. Note that these two edges do not have a common vertex.

figure b

3 Completely regular clique graphs

In this section, we prove Theorem 3. We need the following result.

Proposition 7

(A. Neumaier [13, Theorem 4.1])

Let Γ=(X,R) be a distance-regular graph and let C be a nonempty subset of X with covering radius t=t(C). For i=0,1,…,t, let μ i =|Γ i (x)∩C| and λ i =|Γ i+1(x)∩C| when x∈Γ i (C).

  1. (i)

    C is completely regular if and only if μ i and λ i are independent of the choice of x∈Γ i (C).

  2. (ii)

    Suppose C is completely regular with the parameters γ i ,α i ,β i with i∈{1,2,…,t}. Then

    $$\begin{aligned} \gamma_i &= \frac{\mu_i c_i}{\mu_{i-1}}, \qquad \alpha_i = a_i + \frac {\lambda_ic_{i+1}}{\mu_i} - \frac{\lambda_{i-1}c_i}{\mu_{i-1}}, \quad \textit{and} \\ \beta_i &= b_i - \frac{(\mu_i - \mu_{i-1} - \lambda_{i-1})c_i}{\mu _{i-1}} - \frac{\lambda_ic_{i+1}}{\mu_i}. \end{aligned}$$

Proposition 8

Let \(\widetilde{\varGamma} = (X\cup Y, \widetilde{R})\) be a distance-semiregular graph on X with parameters \(b^{X}_{i}, c^{X}_{i}\) with i=0,1,…,d X. Let Γ be the bipartite half of \(\widetilde{\varGamma}\) on X. For y∈Y, write \(C_{y} = \widetilde{\varGamma}(y) \subset X\), and set \(\mathcal{C}= \{C_{y} \mid y\in Y\}\). Then the following hold.

  1. (i)

    Γ is distance-regular.

  2. (ii)

    Each element \(C\in\mathcal{C}\) is a clique and a completely regular code in Γ.

  3. (iii)

    Each edge in Γ is contained in \(c_{2}^{X}\) members of \(\mathcal{C}\). In particular, Γ=(X,R) is a completely regular clique graph of parameters \((b^{Y}_{0}-1, c^{X}_{2})\).

Proof

(i) This is clear. See [14].

(ii) Since Γ is a distance-2-graph of \(\widetilde{\varGamma}\), each C y is a clique in Γ. Let \(C = C_{y}\in\mathcal{C}\). We apply Proposition 7 to show that C is completely regular. Let x∈X with ∂ Γ (x,C)=i. Then \(\partial _{\widetilde {\varGamma}}(x,y) = 2i+1\). Hence

$$\mu_i(x) = \bigl|\varGamma_i(x)\cap C\bigr| = \bigl|\widetilde{ \varGamma}_{2i}(x)\cap \widetilde{\varGamma}(y)\bigr| = c^X_{2i+1}. $$

Therefore μ i (x) does not depend on the choice of x∈Γ i (C). Since C is a clique, λ i (x)=|C|−μ i (x) and λ i (x) does not depend on the choice of x∈Γ i (C) either.

(iii) For each edge {x 1,x 2} of Γ there exist \(c^{X}_{2}\) vertices y∈Y such that {x 1,x 2}⊂C y . Thus we have the assertions. □

Let Γ=(X,R) be a completely regular clique graph with parameters (s,c) with respect to \(\mathcal{C}\). The incidence graph of Γ is a bipartite graph \(\widetilde{\varGamma} = (X\cup Y, \widetilde {R})\) with vertex set X∪Y, where \(Y = \mathcal{C}\), and edge set \(\widetilde{R} = \{(x,y)\mid x\in X, y\in Y \mbox{ such that } x\in y\}\). Let c i and b i be parameters of Γ if they exist. Let \(c_{i}^{X}\) and \(b_{i}^{X}\) denote the parameters of \(\widetilde{\varGamma}\) when the base vertex is in X and they exist. We define \(c_{i}^{Y}\) and \(b_{i}^{Y}\) similarly.

Lemma 9

Let \(\widetilde{\varGamma} = (X\cup Y, \widetilde{R})\) be the incidence graph of a completely regular clique graph Γ=(X,R) with parameters (s,c) with respect to \(\mathcal{C}\). Then the following hold.

  1. (i)

    \(\widetilde{\varGamma}\) is biregular of valencies \((b^{X}_{0}, b^{Y}_{0})\), where \(b^{X}_{0} = (\beta_{0}+s)c/s\) and \(b^{Y}_{0} = s+1\). Moreover, \(c^{X}_{1} = 1, c^{X}_{2} = c, c^{X}_{3} = \gamma_{1}\), \(b^{X}_{1} = s\), \(b^{X}_{2} = \beta_{0}c/s\), \(b^{X}_{3} = s+1-\gamma_{1}\), and \(b^{Y}_{1} = b^{X}_{0}-1\).

  2. (ii)

    Γ is edge regular with a 1=(s−1)+β 0(γ 1−1)/s.

  3. (iii)

    If d(Γ)>1, then Γ is K 2,1,1-free if and only if γ 1=1 if and only if a 1=s−1. In this case c=1, and each member of \(\mathcal{C}\) is a maximal clique.

Proof

(i) By Lemma 4(ii), Γ is regular of valency k=b 0=α 0+β 0. By definition, we have \(b^{Y}_{0} = s+1\), \(c^{X}_{1} = 1\), \(b^{X}_{1} = b^{Y}_{0}-c^{X}_{1} = s \geq1\), \(c^{X}_{2} = c\) and \(c^{X}_{3} = \gamma_{1}\). We show that \(b^{X}_{0}\) exists.

Let x∈X and let

$$S = \bigl\{ \bigl(x',y\bigr)\in X\times Y\mid\partial_{\tilde{\varGamma}} \bigl(x,x'\bigr) = 2, \; \partial_{\tilde{\varGamma}}(x,y) = \partial_{\tilde{\varGamma}}\bigl(y,x'\bigr) = 1\bigr\} . $$

By counting the cardinality of S, we have \(|S| = |\widetilde{\varGamma }(x)|b^{X}_{1} = b_{0}c^{X}_{2}\). Hence \(b^{X}_{0}\) exists and \(b^{X}_{0} = b_{0}c^{X}_{2}/b^{X}_{1} = kc/s\).

Hence \(\widetilde{\varGamma}\) is biregular. Therefore \(b_{i}^{X}\) exists if and only if \(c^{X}_{i}\) exists, \(b^{X}_{2i}+c^{X}_{2i} = b^{X}_{0}\), \(b^{X}_{2i-1}+c^{X}_{2i-1} = b^{Y}_{0}\). Since α 0=s,

$$b_2^X = b^X_0-c_2^X = kc/s - c = c(k-s)/s = \beta_0c/s. $$

The rest follow immediately.

(ii) Since \(b_{0} = b^{X}_{0}b^{X}_{1}/c^{X}_{2}\) and \(b_{1} = b^{X}_{2}b^{X}_{3}/c^{X}_{2} = \beta _{0}(s+1-\gamma_{1})/s\), we have

$$a_1 = b_0 - c_1 - b_1 = \beta_0+s - 1 - \beta_0(s+1-\gamma_1)/s = (s-1)+\beta_0(\gamma_1-1)/s. $$

(iii) We first prove that Γ is K 2,1,1-free if and only if γ 1=1. Suppose Γ is K 2,1,1-free. Let y∈X, x,z∈Γ(y) with ∂(x,z)=2 and \(\{x,y\}\subset C\in \mathcal{C} \). Since Γ is K 2,1,1-free, γ 1=|Γ(z)∩C|=1. Conversely, suppose γ 1=1. If x,y,z,w form a K 2,1,1 with ∂(z,w)=2 and \(\{x,y\}\subset C\in\mathcal{C}\), then either z or w is not in C. This contradicts γ 1=1.

By (ii) it is clear that γ 1=1 is equivalent to a 1=s−1.

Assume these three equivalent conditions. Then clearly c=1 and each member of \(\mathcal{C}\) is a maximal clique. This proves (iii). □

Next we show that the incidence graph of a distance-regular completely regular clique graph is distance-semiregular.

Proposition 10

Let Γ=(X,R) be a distance-regular graph of valency k and diameter D. Suppose Γ is a completely regular clique graph with parameters (s,c) with respect to \(\mathcal{C}\). Let t be the uniquely determined covering radius and γ i ,α i ,β i with i=0,1,…,t the parameters of completely regular codes in \(\mathcal{C}\). Let \(\widetilde{\varGamma} = (X\cup Y, \widetilde{R})\) be its incidence graph with \(Y = \mathcal{C}\). Then the following hold.

  1. (i)

    \(\widetilde{\varGamma}\) is distance-semiregular on X and Γ is a bipartite half of \(\widetilde{\varGamma}\) on X.

  2. (ii)

    The diameter \(d(\widetilde{\varGamma}) = 2D\) if t=t(C)=D−1 and \(d(\widetilde{\varGamma}) = 2D+1\) if t=t(C)=D and the parameters of \(\widetilde{\varGamma}\) are as follows.

    $$\begin{aligned} b_0^Y &= s+1, \qquad b^X_0 = kc/s,\qquad c^X_1 = 1, \qquad c^X_2 = c, \\ c^X_{2i+1} &=\frac{\gamma_1\gamma_2\cdots\gamma_i}{c_1c_2\cdots c_i}, \qquad c^X_{2j} = \frac{c_1c_1\cdots c_{j-1}c_{j}c}{\gamma_1\gamma_2\cdots \gamma_{j-1}}, \quad \textit{and} \\ b^X_{2i+1} &= s+1-c^X_{2i+1}, \qquad b^X_{2j} = kc/s-c^X_{2j} \end{aligned}$$

    for i,j with \(0\leq2i+1, 2j \leq d(\widetilde{\varGamma})\).

Proof

Let μ i with i=0,1,…,t be the numbers defined in Proposition 7.

(i) For y∈Y, a subset \(\widetilde{\varGamma}(y)\) of X forms a clique of Γ in \(\mathcal{C}\) by definition. We write \(C_{y} =\widetilde {\varGamma}(y)\). Since \(\widetilde{\varGamma}\) is biregular by Lemma 9 (i), \(b_{i}^{X}\) exists if and only if \(c^{X}_{i}\) exists, and \(b^{X}_{2i}+c^{X}_{2i} = b^{X}_{0}\) and \(b^{X}_{2i-1}+c^{X}_{2i-1} = b^{Y}_{0}\). Moreover, if \(b^{X}_{0}, b^{X}_{1}, \ldots, b^{X}_{i-1}, c^{X}_{1}, c^{X}_{2}, \ldots, c^{X}_{i}\) exist, \(k^{X}_{i} \!= \!|\widetilde{\varGamma}_{i}(x)|\! =\! (b^{X}_{0}b^{X}_{1}\cdots b^{X}_{i-1})/(c^{X}_{1}c^{X}_{2} \cdots c^{X}_{i})\) does not depend on the choice of x∈X.

Let y∈Y and x∈Γ i (C y ). Then \(\partial_{\tilde{\varGamma }}(x,y) = 2i+1\) and

$$c_{2i+1}(x,y) = \bigl|\widetilde{\varGamma}_{2i}(x) \cap\widetilde{ \varGamma}(y)\bigr| = \bigl|\varGamma_i(x)\cap C_y\bigr| = \mu_i. $$

Therefore \(c^{X}_{2i+1}\) and hence \(b^{X}_{2i+1}\) exists for all i.

Now suppose \(b^{X}_{0}, b^{X}_{1}, \ldots, b^{X}_{2i+1}, c^{X}_{0}, c^{X}_{1}, \ldots, c^{X}_{2i+1}\) exist for i≥1. For x′∈Γ i+1(x), by counting the cardinality of the set

$$\begin{aligned} S = & \bigl\{ \bigl(x'',y\bigr) \bigm| y \in\widetilde{ \varGamma}_{2i+1}(x)\cap \widetilde {\varGamma}\bigl(x' \bigr), \; x''\in\widetilde{\varGamma}_{2i}(x) \cap\widetilde {\varGamma }(y)\bigr\} \\ = & \bigl\{ \bigl(x'',y\bigr) \bigm| x''\in{\varGamma}_{i}(x)\cap{\varGamma} \bigl(x'\bigr), \; y \in \widetilde{\varGamma}\bigl(x' \bigr)\cap\widetilde{\varGamma}\bigl(x''\bigr)\bigr\} \end{aligned}$$

we have \(c_{i+1}c^{X}_{2} = c^{X}_{2i+1}|\widetilde{\varGamma}_{2i+1}(x)\cap \widetilde{\varGamma}(x')| = c^{X}_{2i+1}c_{2i+2}(x,x')\). Thus \(c^{X}_{2i+2}\) exists and

$$c^X_{2i+2} = c_{i+1}c^X_2/c^X_{2i+1}. $$

By induction, we have that \(\widetilde{\varGamma}\) is distance-semiregular on X.

(ii) Note that μ 0=1. Now by Proposition 7,

$$c^X_{2i+1} = \mu_i = \frac{\gamma_1\gamma_2\cdots\gamma _i}{c_1c_2\cdots c_i}, \quad \mbox{and}\quad c^X_{2i+2} = \frac{c_{i+1}c^X_2}{c^X_{2i+1}} = \frac {c_1c_1\cdots c_ic_{i+1}c}{\gamma_1\gamma_2\cdots\gamma_i}. $$

Thus we have the assertion. □

Theorem 3 now follows directly from Propositions 8 and 10.

4 Notes

It is useful to define the notion of completely regular clique graphs in terms of an incidence structure.

Definition 2

Let \(\mathcal{I}= (X, Y, I)\) be an incidence structure, where X and Y are finite set and I a relation on X×Y. Let Γ=(X,R) be the collinearity graph of \(\mathcal{I}\) with vertex set X and edge set \(R =\{ \{x,x'\}\subset X \mid x\neq x' \mbox{ and there exists $y\in Y$ such that }x I y \mbox{ and }x' I y\}\). Then \(\mathcal{I}\) is said to be a CRC geometry with parameters (s,c), if the following are satisfied.

  1. (i)

    For each y∈Y, let C y ={x∈X∣xIy}. Then C y is a completely regular code of Γ of size s+1≥2.

  2. (ii)

    For each distinct x,x′∈X, |{y∈Y∣xIy and x′Iy}|∈{0,c} and c≥1.

Let Γ=(X,R) be one of the Johnson graphs or the Grassmann graphs of diameter D. Then for each i∈{0,1,…,D−1} and x,y∈X with ∂(x,y)=i we can choose a geodetically closed completely regular code C x,y of diameter i containing x and y. Define \(\mathcal{C}_{i} = \{C_{x,y} \mid x, y\in X \mbox{ with }\partial (x,y) = i\}\) and \(\mathcal{C}_{i+1}\) similarly. Then the pair \((\mathcal{C}_{i}, \mathcal{C} _{i+1})\) defines a CRC geometry when incidence is defined by inclusion.

We conclude this paper by proposing three problems related to completely regular clique graphs.

Problem 1

Prove or disprove that completely regular clique graphs are distance-regular.

Problem 2

Classify completely regular clique graphs with respect to \(\mathcal {C}\) such that there is a nonempty collection \(\mathcal{C}'\) of completely regular codes of width two such that \((\mathcal{C}, \mathcal{C}', \subset)\) is a CRC geometry when incidence is defined by inclusion.

Problem 3

Characterize connected graphs Γ=(X,R) with the following properties.

  1. (i)

    There is a nonempty collection \(\mathcal{C}\) of cliques of size s+1 and that for each \(C\in\mathcal{C}\) and x∈C, \(\mathcal{T}(C)\boldsymbol{c}_{x}\) is a thin irreducible \(\mathcal{T}(C)\)-module, where \(\boldsymbol{c}_{x} = \hat{x} - \frac {1}{|C|}\sum_{y\in C}\hat{y} \in\boldsymbol{C}^{X}\). Here \(\hat{z}\) is a unit vector in C X corresponding to a vertex z.

  2. (ii)

    Each edge is contained in exactly c members of \(\mathcal{C} \) and c≥1.