Combinatorial vs. Algebraic Characterizations of Completely Pseudo-Regular Codes

Given a simple connected graph Γ and a subset of its vertices C, the pseudodistance-regularity around C generalizes, for not necessarily regular graphs, the notion of completely regular code. We then say that C is a completely pseudoregular code. Up to now, most of the characterizations of pseudo-distance-regularity has been derived from a combinatorial definition. In this paper we propose an algebraic (Terwilliger-like) approach to this notion, showing its equivalence with the combinatorial one. This allows us to give new proofs of known results, and also to obtain new characterizations which do not depend on the so-called C-spectrum of Γ, but only on the positive eigenvector of its adjacency matrix. Along the way, we also obtain some new results relating the local spectra of a vertex set and its antipodal. As a consequence of our study, we obtain a new characterization of a completely regular code C, in terms of the number of walks in Γ with an endvertex in C. 1 Preliminaries Pseudo-distance-regularity is a natural generalization of distance-regularity which extends this notion to not necessarily regular graphs. The key point of this generalization relays ∗Research supported by the “Ministerio de Ciencia e Innovación” (Spain) with the European Regional Development Fund under projects MTM2008-06620-C03-01 and by the Catalan Research Council under project 2005SGR00256. the electronic journal of combinatorics 17 (2010), #R37 1 on defining an adequate weight for each vertex in such a way that we obtain a “regularized” graph. Since its introduction in [7], the study of pseudo-distance-regularity produced several interesting results, specially in the area of quasi-spectral characterizations of distance-regularity [4, 7] and completely regular codes [5, 6]. This study was based on the combinatorial definition of pseudo-distance-regularity around a vertex, which comes up naturally from the notion of distance-regularity around a vertex. Among the variety of techniques used in these works, two concepts stand out: the local spectrum (of a single vertex or a subset of vertices) and certain families of orthogonal polynomials. Our work in this paper is motivated by the connection existing between pseudodistance-regularity and the study developed by Terwilliger [11] in the context of association schemes. In his work, he introduced the subconstituent algebra (also known as Terwilliger algebra) with respect to a vertex of a graph and defined the notion of thin module in this algebra. As commented by the third and fourth authors in [3, 5], the concept of pseudo-distance-regularity around a vertex i is equivalent to the thin character of the minimum module containing its characteristic vector ei. The aim of this paper is to extend this parallelism from a single vertex to a set of vertices. The plan of the paper is as follows. In the rest of this section we first give some notation on graphs and their spectra. In Section 2 we introduce the local spectrum of a vertex set, discussing some of its properties. Special attention is paid to the relation between the local spectra of two antipodal subsets of vertices. Section 3 is devoted to explain the concept of pseudo-distance-regularity around a vertex set, in combinatorial sense, and to review some of its known quasi-spectral characterizations. In the case of regular graphs, this concept coincides with that of a completely regular code. According to this fact, we say that a set of vertices satisfying this property is a completely pseudo-regular code. Our main results are in Section 4, where we extend the (algebraic) definition of Terwilliger to a set of vertices in any graph, and prove its equivalence with the combinatorial approach. This allows us to give new proofs of known results, and also to obtain new characterizations which do not depend on the so-called C-local spectrum, but only on the positive eigenvector of the adjacency matrix. As a consequence, we obtain a new characterization of a completely regular code C, in terms of the number of walks having an endvertex in C. Throughout this paper Γ = (V,E) stands for a simple connected graph with vertex set V = {1, 2, . . . , n} and V denotes the space of the formal linear combinations of its vertices. The adjacencies in Γ, say {i, j} ∈ E, are denoted by i ∼ j and Γk(i) = {j | ∂(i, j) = k} represents the set of vertices at distance k from i, where ∂(·, ·) is the distance function in Γ. For simplicity we will write Γ(i) instead of Γ1(i). Every vertex i is associated to the i-th unitary (or characteristic) vector ei ∈ R n and, consequently, V is identified with R. With this identification in mind, the adjacency matrix A of Γ can be seen as the matrix of an endomorphism in V with respect to the basis {ei}i∈V . The set of different eigenvalues of A is denoted by evΓ := {λ0, λ1, . . . , λd}, where λ0 > λ1 > · · · > λd, and the spectrum of Γ is defined by spΓ := spA = {λ m(λ0) 0 , λ m(λ1) 1 , · · · , λ m(λd) d }, where m(λl) stands for the multiplicity of the eigenvalue λl. From the Perron-Frobenius the electronic journal of combinatorics 17 (2010), #R37 2 theorem for nonnegative matrices, we have that λ0 > |λd| and equality is attained if and only if Γ is a bipartite graph; see e.g. [1]. Moreover, m(λ0) = 1 and every non-null vector of Ker(A − λ0I) has all its components either positive or negative. We denote by ν ∈ Ker(A − λ0I) the unique positive eigenvector with minimum component equal to one. Let us remark that in the case of δ-regular graphs we have that λ0 = δ and the vector ν turns out to be the all-1 vector j. Note that V is a module over the quotient ring R[x]/I, where I is the ideal generated by the polynomial Z = ∏d l=0(x− λl), which vanishes in A, with product defined by pu := p(A)u for every p ∈ R[x]/I and u ∈ V. Recall that, for every 0 6 l 6 d, the orthogonal projection El of V onto the eigenspace El = Ker(A− λlI) can be written as Elu = Zlu, u ∈ V, where Zl = (−1) πl ∏ 06h6d(h 6=l)(x− λl) and πl := ∏ 06h6d(h 6=l) |λh − λl|. 2 The local spectrum of a vertex set and its antipodal Given a nonempty set C of vertices of Γ, we consider the map ρ : P(V ) → V defined by ρ∅ = 0 and ρC = ∑ i∈C νiei for C 6= ∅ and denote by eC the normalized vector ρC/‖ρC‖. If eC = zC(λ0) + zC(λ1) + · · · + zC(λd) is the spectral decomposition of eC; that is zC(λl) = EleC ∈ El, 0 6 l 6 d, the C-multiplicity (or C-local multiplicity) of the eigenvalue λl is defined by mC(λl) = ‖zC(λl)‖ . Note that, since zC(λ0) = E0eC = 1 ‖ρC‖ 〈ρC,ν〉 ‖ν‖2 ν = 1 ‖ρC‖ ∑ i∈C νi νi ‖ν‖2 ν = ‖ρC‖ ‖ν‖2 ν, we get mC(λ0) = ‖ρC‖ ‖ν‖2 . Then, if μ0(= λ0), μ1, . . . , μdC are the eigenvalues with non-zero C-multiplicity, the C-spectrum (or C-local spectrum) is defined by spC Γ := {μ mC(μ0) 0 , μ mC(μ1) 1 , . . . , μ mC(μdC ) dC }, with μ0 > μ1 > · · · > μdC , and the set of different eigenvalues of C is denoted by evC Γ := {μ0, μ1, . . . , μdC}. Note that, since eC is unitary, we have ∑dC l=0mC(λl) = 1 or, equivalently, the vector mC = (‖zC(μ0)‖, ‖zC(μ1)‖, . . . , ‖zC(μdC )‖) ∈ R C, is also unitary. As we have done for the spectrum of Γ, in order to simplify notation we introduce the moment-like parameters πl(C) := ∏ 06h6dC(h 6=l) |μh − μl| (0 6 l 6 dC). the electronic journal of combinatorics 17 (2010), #R37 3 The set Γk(C) = {v ∈ V | ∂(v, C) = k} of vertices at distance k from C is denoted by Ck. Thus, if C has eccentricity εC, C0(= C), C1, . . . , CεC is a partition of V . We denote by C the set CεC of vertices at maximum distance from C, and we refer to it as its antipodal set. If there is no possible confusion, we will write D = C. The polynomial ZC = ∏dC l=0(x − μl) is the monic polynomial with minimum degree such that ZCeC = 0, and the polynomial HC = ‖ν‖ π0(C)‖ρC‖ dC ∏ l=1 (x− μl) (1) satisfies HCν = HC(λ0)ν = ‖ν‖ ‖ρC‖2ν. What is more, HC is the unique polynomial of degree at most dC satisfying HCρC = ‖ρC‖ ‖ν‖2 HCν = ν (2) and so, inspired by Hoffman [8], it is named the C-local Hoffman polynomial. This allows us to conclude that the eccentricity of C and the number of C-local eigenvalues are related by εC 6 dC; see [5]. In case of equality, εC = dC, we say that C is extremal. Proposition 2.1 Let C be an extremal set and let D be its antipodal set. Then, evC Γ ⊂ evD Γ and the C-multiplicities and D-multiplicities satisfy mC(μl)mD(μl) > π 0(C) π l (C) ‖ρC‖‖ρD‖ ‖ν‖4 for all μl ∈ evC Γ, where equality is equivalent to the linear dependence of the vectors zC(μl) and zD(μl). Proof. Consider the interpolating polynomials associated with the local spectrum of C: ZC l = (−1) πl(C) ∏ 06h6dC (h 6=l) (x− μh) (0 6 l 6 dC), (3) verifying Z l (μh) = δlh. Since both Z C l andHC have degree dC and their leading coefficients are, respectively, (−1) l πl(C) and ‖ν‖ 2 π0(C)‖ρC‖ , the polynomial T = π0(C) ‖ρC‖ ‖ν‖2 HC − (−1) πl(C)Z C l has degree less than dC. The extremal character of C gives 〈ρC,ZC l ρD〉 = 〈Z C l ρC,ρD〉 = (−1) πl(C) 〈xCρC,ρD〉 6 = 0. In particular, Z l ρD 6= 0. Moreover, if μl ∈ evC Γ, 〈ρC,ZC l ρD〉 = 〈 ρC, ∑d h=0Z C l (λh)EhρD 〉