On the local spectra of the subconstituents of a vertex set and completely pseudo-regular codes

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On the local spectra of the subconstituents of a vertex set and completely pseudo-regular codes✩ M. Cámara, J. Fàbrega, M.A. Fiol ∗ , E. Garriga Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, BarcelonaTech, Jordi Girona 1-3, 08034 Barcelona, Catalonia, Spain

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Article history: Received 12 November 2012 Received in revised form 23 August 2013 Accepted 27 September 2013 Available online xxxx Keywords: Pseudo-distance-regularity Local spectrum Subconstituents Predistance polynomials Completely regular code

abstract In this paper we study the relation between the local spectrum of a vertex set C and the local spectra of its subconstituents. In particular, it is shown that, when C is a completely regular code, such spectra are uniquely determined by the local spectra of C . Moreover, we obtain a new characterization for completely pseudo-regular codes, and consequently for completely regular codes, in terms of the relation between the local spectrum of an extremal set of vertices and the local spectrum of its antipodal set. We also present a new proof of the version of the spectral excess theorem for extremal sets of vertices. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The notion of local spectrum (see Section 2) was first introduced by Fiol, Garriga, and Yebra [14] for a single vertex of a graph. In that paper, such a concept was used to obtain several quasi-spectral characterizations of local (pseudo)-distanceregularity. In the study of pseudo-distance-regularity around a set of vertices, first done by Fiol and Garriga [12], which particularizes to that of completely regular codes when the graph is regular, the local spectrum is generalized to a set of vertices C (and, thus, it is called the C -local spectrum). As commented in the same paper, when we study the graph from a ‘base’ vertex subset C , its local spectrum plays a role similar to the one played by the (standard) spectrum for studying the whole graph. In this work we are interested in the study of the relation between the local spectrum of a vertex set and the local spectra of the elements of the distance partition associated with it (also known as its subconstituents). Thus, Section 2 is devoted to define the local spectrum of a subset of vertices in a graph. In Section 3 completely pseudo-regular codes are introduced and we discuss some known results of special interest. Our main results can be found in Sections 4 and 5, where we give sufficient conditions implying a tight relation between the local spectrum of a set of vertices and that of each of its subconstituents. As a consequence, we obtain a new characterization of completely (pseudo-)regular codes. In the way, we also obtain some information about the structure of the local spectrum of the subconstituents associated with a completely pseudo-regular code and we give a new proof of a result by Fiol and Garriga [12,13], which can be seen as the spectral excess theorem, due to the same authors [11], for sets of vertices.

✩ This research was supported by the Spanish Council under project MTM2011-28800-C02-01 and by the Catalan Research Council under project 2009SGR01387. ∗ Corresponding author. Tel.: +34 934015993; fax: +34 934015981. E-mail addresses: [email protected] (M. Cámara), [email protected] (J. Fàbrega), [email protected] (M.A. Fiol), [email protected] (E. Garriga).

0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.09.018

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Before going into our study, let us first give some notation. In this paper Γ = (V , E ) stands for a simple connected graph with a vertex set V = {1, 2, . . . , n}. Each vertex i ∈ V is identified with the i-th unit coordinate (column) vector ei and V ∼ = Rn denotes the vector space of formal linear combinations of its vertices. The distance function in Γ is denoted by dist(·, ·). Given a set of vertices C ⊂ V , the distance from a vertex i to C is given by the expression dist(i, C ) = min{dist(i, j)|j ∈ C }. We denote by εC = maxi∈V dist(i, C ) the eccentricity of C . Notice that, in the context of coding theory, the parameter εC corresponds to the covering radius of the code C . As usual, A stands for the adjacency matrix of Γ , with a set of different eigenvalues evΓ = evA = {λ0 , λ1 , . . . , λd }, where λ0 > λ1 > · · · > λd . The spectrum of Γ is m(λ0 )

spΓ = spA = {λ0

m(λ1 )

, λ1

m(λd )

, . . . , λd

},

where m(λl ) is the multiplicity of the eigenvalue λl . We denote by El = ker(A − λl I ) the eigenspace of A corresponding to λl . Recall that, since Γ is connected, E0 is one-dimensional, and all its elements are eigenvectors having all its components either positive or negative (see, for instance, Biggs [1] or Cvetković, Doob, and Sachs [6]). Denote by ν = (ν1 , ν2 , . . . , νn ) ∈ E0 the unique positive eigenvector of Γ with minimum component equal to 1. Note that V is a module over the quotient ring R[x]/(Z ), where (Z ) is the ideal generated by the minimal polynomial of d A, Z = l=0 (x − λl ), with product defined by pu = p(A)u

for every p ∈ R[x]/(Z ) and u ∈ V .

With this notation, let us remark that the orthogonal projection El of V onto the eigenspace El corresponds to El u = Zl u,

u ∈ V,

where Zl , l = 0, 1, . . . , d, is the Lagrange interpolating polynomial satisfying Zl (λh ) = δlh , that is, Zl =

(−1)l πl



(x − λh ),

0≤h≤d, h̸=l

with πl being the moment-like parameter given by πl =



0≤h≤d, h̸=l

|λl − λh |.

2. Local spectrum of a vertex set Given a set of vertices C of Γ , define the map ρ : P (V ) → V by ρ∅ = 0 and ρC = i∈C νi ei for C ̸= ∅. Consider the spectral decomposition of the unit vector eC = ρC /∥ρC ∥ = zC (λ0 ) + zC (λ1 ) + · · · + zC (λd ), that is zC (λl ) = El eC ∈ El , 0 ≤ l ≤ d. The C -local multiplicity of the eigenvalue λl is defined by



mC (λl ) = ⟨El eC , eC ⟩ = ∥zC (λl )∥2 . If µ0 > µ1 > · · · > µdC are the eigenvalues of Γ with nonzero C -local multiplicity, the C -local spectrum of Γ is defined by m (µ0 )

spC Γ = {µ0 C

m (µ1 )

, µ1 C

mC (µd )

, . . . , µdC

C

},

and we denote by evC Γ = {µ0 , µ1 , . . . , µdC }, µ0 > µ1 > · · · > µdC , the set of different eigenvalues in the C -local spec⟨e ,ν⟩

∥ρC ∥2

C trum. Let us remark that, as E0 eC = ∥ν∥ 2 ν = ∥ν∥2 ν, we have mC (λ0 ) = ∥ν∥2 ̸= 0, and hence µ0 = λ0 . The parameter dC is called the dual degree of C and it provides an upper bound for the eccentricity of the vertex set, εC ≤ dC (see [12]). When the equality is attained we say that C is extremal. Notice that, if Γ is regular, then ν = j is the all-1 vector, and ρC is the characteristic vector of C , χ = χC . Then, the above concepts reduce to standard Delsarte’s theory [7,8] (see also Brouwer, Cohen and Neumaier [2, Section 2.5]). For instance, in this context, the sequence of C -local multiplicities mC (λ0 ), . . . , mC (λd ) is the so-called MacWilliams transform of the vector eC . Consider the idempotents ElC , 0 ≤ l ≤ dC , corresponding to the members of the C -local spectrum, that is ElC is the  projection of VC = we have done for the standard spectrum of the λh ∈evC Γ Eh onto the eigenspace corresponding to µl . As  graph, we define for each µl ∈ evC Γ the moment-like parameter πl (C ) = 0≤h≤dC , h̸=l |µl − µh | and consider the Lagrange interpolation polynomial

ZlC =

(−1)l πl (C )



( x − µh )

∥ρC ∥

(1)

0≤h≤dC , h̸=l

which gives ZlC (A) = ElC . 3. Completely pseudo-regular codes In this section we review some known results on completely pseudo-regular codes. These results are formulated in terms of C -local pseudo-distance-regularity, which extends the notion of local distance-regularity from single vertices to subsets

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of vertices and from regular to non-necessarily regular graphs. Completely pseudo-regular codes were introduced in [12] with the aim of generalizing both local distance-regularity and completely regular codes [1,7]. Let us consider the distance partition of a vertex set C ⊂ V of Γ , given by the sets Ck = {i ∈ V |dist(i, C ) = k}, k = 0, 1, . . . , εC , which are known as the subconstituents associated with C . Consider also the functions a, b, c : V −→ [0, λ0 ] acting on a vertex i ∈ Ck as follows: c (i) =

 0

1

 νi 1

(k = 0); 

νj

(1 ≤ k ≤ εC ).

j∈Γ (i)∩Ck−1



νj (0 ≤ k ≤ εC ). νi j∈Γ (i)∩C k   1   νj (0 ≤ k ≤ εC − 1); b(i) = νi j∈Γ (i)∩Ck+1   0 (k = εC ). a(i) =

We say that C is a completely pseudo-regular code (or that Γ is C -local pseudo-distance-regular) when the values of c , a, and b do not depend on the chosen vertex i ∈ Ck but only on k; that is, when the distance partition with respect to C is pseudo-regular (see [9]). If this is the case, we denote by ck , ak and bk the values of these three functions on the vertices of Ck , k = 0, 1, . . . , εC , and refer to them as the C -local pseudo-intersection numbers. Remark that, since ν ∈ E0 , the sum of these three functions is constant over all the vertices of Γ : c (i) + a(i) + b(i) =

1 

νi

j∈Γ (i)

νj =

1

νi

(Aν)i = λ0 for all i ∈ V .

Thus, in a sense, we can say that the weight νi , given to every vertex i, regularizes the graph. Note also that when Γ is a regular graph we have that ν = j, and the definition of a completely pseudo-regular code particularizes to that of a completely regular code, first considered by Biggs [1] and Delsarte [7]. Completely regular codes, also known as completely regular sets, have been studied by the different authors, such as Courteau and Montpetit [5], Godsil [15], Martin [17], and Neumaier [18], among others. 3.1. Some characterizations of completely pseudo-regular codes As established by Cámara, Fàbrega, Fiol, and Garriga [4], one can consider two approaches to completely pseudo-regular codes. One is based on the combinatorial definition given in the previous section and the other relies upon the study of the Terwilliger algebra [19] associated with a vertex set. Although the results presented here can be obtained from both points of view, we will focus the attention on the combinatorial approach, which makes more evident the key role of the local spectrum.  From now on, all the polynomials must be considered in the quotient ring R[x]/(ZC ), where ZC = µl ∈evC Γ (x − µl ). We begin by defining the C -local scalar product as follows:

⟨p, q⟩C = ⟨peC , qeC ⟩ =

dC 

mC (µl )p(µl )q(µl ).

l =0

Then, we consider the family of C -local predistance polynomials, (pk )0≤k≤dC , which is the unique orthogonal system with respect to the C -local scalar product, such that deg pk = k and ⟨pk , ph ⟩C = δkh pk (µ0 ), 0 ≤ k ≤ dC ; see Càmara, Fàbrega, Fiol, and Garriga [3]. In [12] one can find some quasi-spectral characterizations of completely pseudo-regular codes, which are based on these polynomials. In particular, the following two theorems are of special interest in our work. Theorem 1. Let Γ = (V , E ) be a graph and let C ⊂ V have eccentricity εC . Then, C is a completely pseudo-regular code if and only if the C -local (pre)distance polynomials satisfy ρCk = pk ρC , k = 0, 1, . . . , εC . The following result establishes that, if we assume that C is extremal, it is enough to check the condition of Theorem 1 for the set of vertices at maximum distance CεC . We refer to this set as the antipodal set of C and, if there is no possible confusion, we set D = CεC . Theorem 2. Let Γ = (V , E ) be a graph and let C be an extremal set with eccentricity εC = dC . Then, C is a completely pseudoregular code if and only if there exists a polynomial p ∈ RdC [x] such that pρC = ρD, in which case p is the C -local predistance polynomial pdC . This last result points out the relevance of the antipodal set in the study of completely pseudo-regular codes. In fact, it is known from [12] that a set of vertices is a completely pseudo-regular code if and only if its antipodal set is. This symmetry is illustrated in Fig. 1 where the pk ’s are the D-local predistance polynomials.

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Fig. 1. Antipodal completely pseudo-regular codes and their distance partition.

4. Spectra of the subconstituents As showed in [4], for an extremal set of vertices C with antipodal set D, the relation between the local spectra of C and D is tight and can be expressed in terms of the following proposition. We include the proof for the sake of completeness. Proposition 1. Let C be an extremal set and let D be its antipodal set. Then, evC Γ ⊂ evD Γ and the C -local multiplicities and D-local multiplicities satisfy mC (µl )mD (µl ) ≥

π02 (C ) ∥ρC ∥2 ∥ρD∥2 ∥ν∥4 πl2 (C )

for all µl ∈ evC Γ .

(2)

Moreover: (i) Equality holds in Eq. (2) for an eigenvalue µl if and only if the vectors zC (µl ) and zD (µl ) are linearly dependent. (ii) Equality in Eq. (2) for every µl ∈ evC Γ is equivalent to the existence of a polynomial p ∈ RεC [x] such that:

ρD = pρC + z ,

where z ∈



El .

λl ∈evD Γ \evC Γ

Proof. Consider the C -local Hoffman polynomial (see [4,16]) HC =

dC  ∥ν∥2 C ∥ν∥2 (x − µl ), Z = 0 ∥ρC ∥2 π0 (C )∥ρC ∥2 l=1

which satisfy HC ρC = ν. Since both ZlC , defined in Eq. (1), and HC have degree dC and their leading coefficients are, (−1)l

∥ν∥2

respectively, π (C ) and π (C )∥ρC ∥2 , the polynomial l 0 T = π0 (C )

∥ρC ∥2 HC − (−1)l πl (C )ZlC ∥ν∥2

has degree less than dC = εC . Then, the vectors T eC and eD are orthogonal, giving:

∥ρC ∥2 ⟨HC eC , eD ⟩ − (−1)l πl (C )⟨ZlC eC , eD ⟩ ∥ν∥2 ∥ρC ∥ = π0 (C ) ⟨HC ρC , ρD⟩ − (−1)l πl (C )⟨zC (µl ), eD ⟩ ∥ρD∥ ∥ν∥2 ∥ρC ∥ ∥ρD∥ = π0 (C ) − (−1)l πl (C )⟨zC (µl ), zD (µl )⟩ = 0, ∥ν∥2

⟨T eC , eD ⟩ = π0 (C )

(3)

since ⟨HC ρC , ρD⟩ = ⟨ν, ρD⟩ = ∥ρD∥2 . Therefore, by the Cauchy–Schwarz inequality,

π02 (C ) ∥ρC ∥2 ∥ρD∥2 = ⟨zC (µl ), zD (µl )⟩2 ≤ mC (µl )mD (µl ), ∥ν∥4 πl2 (C ) where equality occurs if and only if zC (µl ) and zD (µl ) are collinear, and (i) is also proved. Moreover, as all the terms involved are positive, mD (µl ) > 0 and hence µl ∈ evD Γ .

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In order to proof (ii), suppose now that the equality holds for every eigenvalue of the C -local spectrum. Then, given

µl ∈ evC Γ , the vectors zD (µl ), zC (µl ) are proportional. More precisely, by Eq. (3), there exist ξl > 0 such that zD (µl ) = D∥ (−1)l ξl zC (µl ). Let p be the unique polynomial in RεC [x] such that p(µl ) = (−1)l ∥ρ ξ for all µl ∈ evC Γ . We have ∥ρC ∥ l El ρD = ∥ρD∥zD (µl ) = (−1)l ∥ρD∥ξl zC (µl ) ∥ρD∥ ξl El ρC = p(µl )El ρC = El pρC . = (−1)l ∥ρC ∥

Thus z = ρD − pρC ∈ λl ∈evD Γ \evC Γ El . Conversely, assuming the existence of p ∈ RεC [x] satisfying ρD = pρC + z, with  z ∈ E , by projecting onto the eigenspace of µl ∈ evC Γ we obtain ∥ρD∥zD (µl ) = p(µl )∥ρC ∥zC (µl ) and, by l λl ∈evD Γ \evC Γ (i), equality in Eq. (2) holds for every µl ∈ evC Γ . 



Although the last result cannot be directly extended to the other subconstituents, it suggests that the existence of the distance polynomials guarantees a tight relation between the local spectra of the subconstituents. The following proposition supports this claim. Proposition 2. Let C be a completely pseudo-regular code in a graph Γ . Denote by Ck , k = 0, 1, . . . , ε(= dC ) the subconstituents associated with C . Then evCk Γ ⊂ evC Γ . Moreover, for each l = 0, 1, . . . , dC , the projections of ρCk and ρC onto the eigenspace El are linearly dependent. Proof. Let (pk )0≤k≤ε be the C -local predistance polynomials. From Theorem 1 we have that ρCk = pk ρC , k = 0, 1, . . . , ε . By projecting onto the eigenspace El we obtain El ρCk = El pk ρC = pk (λl )El ρC , so that El ρCk and El ρC are collinear. Moreover, ρC ρC since eCk = ∥ρCk ∥ and eC = ∥ρC ∥ we obtain k

mCk (λl ) = ∥El eCk ∥2 =

∥El ρCk ∥2 ∥pk (λl )El ρC ∥2 = 2 ∥ρCk ∥ ∥ρCk ∥2

=

∥ρC ∥2 (pk (λl ))2 ∥El eC ∥2 ∥ρCk ∥2

=

∥ρC ∥2 (pk (λl ))2 mC (λl ). ∥ρCk ∥2

(4)

Thus, mCk (λl ) = 0 when either pk (λl ) = 0 or mC (λl ) = 0, and evCk Γ ⊂ evC Γ .



Note that, since for an extremal set we have evC Γ ⊂ evD Γ , the last result shows that, in particular, for a completely pseudo-regular code we have evC Γ = evD Γ , and spC Γ = spD Γ if and only if pdC (µl ) = (−1)l ∥ρD∥/∥ρC ∥ for all µl ∈ evC Γ . Also, notice that, since the C -local predistance polynomials are uniquely determined by the local spectrum of C , the same holds for the local spectrum of each subconstituent Ck . As another by-product of Eq. (4) for k = dC and Proposition 1(ii), we have that, for a completely pseudo-regular code, mC (µl ) =

1 π0 (C ) ∥ρD∥2 ; 2 πl (C ) ∥ν∥ pdC (µl )

(5)

mD (µl ) =

π0 (C ) ∥ρC ∥2 pd (µl ); πl (C ) ∥ν∥2 C

(6)

for all µl ∈ evC Γ . As commented, recall that in this case D is also a completely pseudo-regular code and C is its antipodal set. If we denote by p the D-local predistance polynomial with maximum degree and use the existing symmetry (see Fig. 1) Eq. (5) gives mD (µl ) =

π0 (C ) ∥ρC ∥2 1 for all µl ∈ evC Γ . πl (C ) ∥ν∥2 p(µl )

This jointly with Eq. (6) leads us to the existing relation between the C -local and D-local predistance polynomials of maximum degree: p(µl ) =

1 pdC (µl )

for all µl ∈ evC Γ .

Recall that the C -local predistance polynomials satisfy a three term recurrence (see [3]) and, in particular, for a completely pseudo-regular code with intersection numbers ak , bk and ck , k = 0, 1, . . . , εC (= dC ), we have xpk = bk−1 pk−1 + ak pk + ck+1 pk+1

(0 ≤ k ≤ dC ),

where b−1 = cdC +1 = 0 and bk ck+1 > 0 (see [3]). Thus, for an eigenvalue µl ∈ evC Γ pk+1 (µl ) =

(µl − ak )pk (µl ) − bk−1 pk−1 (µl ) ck+1

,

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and, since evC Γ = evD Γ , Eq. (4) ensures us that in a completely pseudo-regular code there cannot exist k such that µl ̸∈ evCk Γ ∪ evCk+1 Γ . That is, for every k = 0, 1, . . . , εC − 1, evC Γ = evCk Γ ∪ evCk+1 Γ . Remark also that in this case we have that εC1 ≥ dC − 1 and, in general, εCk ≥ max{k, dC − k}. Thus, since dCk ≥ εCk , the dual degree of the k-th subconstituent satisfies the bound dCk ≥ max{k, dC − k}. Consequently, in a completely pseudo-regular code, the C -local predistance polynomial pk vanishes at most at min{k, dC − k} different eigenvalues of the C -local spectrum. 5. Characterizations of completely pseudo-regular codes Some of the results given in the previous section can be used to obtain characterizations of completely pseudo-regular codes. In particular, we get an alternative proof for extremal sets of a result in [12], which can be seen as the version of the spectral excess theorem, due to Fiol and Garriga [11], for sets of vertices (for short proofs of such a theorem, see Van Dam [20] and Fiol, Gago, and Garriga [10]). Theorem 3. Let C be an extremal set and D its antipodal set. Then

∥ρD∥2 1/mC (λ0 )2 π02 (C ) ≤ d , 2 C ∥ρC ∥  2 1/mC (µl )πl (C )

(7)

l =0

and the equality holds if and only if C is a completely pseudo-regular code. Proof. By Proposition 1 we have mD (µl ) ≥

π02 (C ) ∥ρC ∥2 ∥ρD∥2 1 for all µl ∈ evC Γ . ∥ν∥4 mC (µl ) πl2 (C ) ∥ρC ∥2

By adding up for all µl ∈ evC Γ , and using that mC (λ0 ) = ∥ν∥2 , we obtain dC dC d   π02 (C ) ∥ρD∥2 mC (λ0 )2  ≤ m (µ ) ≤ mD (λl ) = ∥eD ∥2 = 1, D l 2 πl (C ) ∥ρC ∥2 mC (µl ) l =0 l =0 l =0

(8)

giving Eq. (7). In the case of equality in Eq. (8) we obtain mD (µl ) = 0 if λl ̸∈ evC Γ , so that evC Γ = evD Γ . Moreover, in this case Proposition 1(ii) applies and there exists a polynomial p ∈ RεC [x] such that pρC = ρD, or, equivalently, C is a completely pseudo-regular code.  When the graph is regular ∥ρD∥2 /∥ρC ∥2 = |D|/|C |, and the above result corresponds to the spectral excess theorem for completely regular codes (see [12]), from which Fiol and Garriga [13] derived an improved version when the graph is distance-regular. Next theorem gives a new approach to completely pseudo-regular codes and shows that, when the considered set of vertices is extremal, the converse of Proposition 2 also holds. Theorem 4. Let C be an extremal set with eccentricity ε = dC and antipodal set D = Cε . Then C is a completely pseudo-regular code if and only if the orthogonal projections of ρC and ρD onto each eigenspace of Γ are collinear. Proof. Proposition 2 guarantees the necessity. Assume now that El ρD and El ρC are collinear for each l = 0, 1, . . . , d. That is, there exist constants αl satisfying El ρD = αl El ρC . Let pε be the unique polynomial of degree ε such that pε (λl ) = αl . Then,

ρD =

d 

El ρ D =

l =0

d 

αl El ρC =

d 

l =0

and the result follows from Theorem 2.

pε (λl )El ρC = pε ρC ,

l=0



In particular, if the underlying graph Γ is regular, Theorem 4 gives a new characterization of completely regular codes. Moreover, if C is a single vertex i, then ρC = ei and ρD = χ = χD . Hence, we conclude that the graph is distance-regular around i if and only if the i-th column of each idempotent El is a multiple of χ. Acknowledgments The authors sincerely acknowledge the referees of this work for helpful comments and suggestions, which led to a significant improvement of the paper.

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