Contents lists available atScienceDirect
Linear Algebra and its Applications
www.elsevier.com/locate/laaA characterization of bipartite distance-regular
graphs
Guang-Siang Lee∗, Chih-wen Weng
Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan, ROC
a r t i c l e i n f o a b s t r a c t
Article history:
Received 9 April 2013 Accepted 15 December 2013 Available online 13 January 2014 Submitted by R. Brualdi MSC: 05E30 05C50 Keywords: Distance-regular graph Distance matrices Predistance polynomials Spectral diameter Spectral excess theorem
It is well-known that the halved graphs of a bipartite distance-regular graph are distance-distance-regular. Examples are given to show that the converse does not hold. Thus, a natural question is to find out when the converse is true. In this paper we give a quasi-spectral characterization of a connected bipartite weighted 2-punctually distance-regular graph whose halved graphs are distance-regular. In the case the spectral diameter is even we show that the graph characterized above is distance-regular.
© 2013 Elsevier Inc. All rights reserved.
1. Introduction
The study of characterizing the graphs whose eigenvalues and/or multiplicities sat-isfy a prescribed identity has a long history. For example, a well-known and real-world applicable theory asserts that a connected graph is bipartite if and only if its largest eigen-value and smallest eigeneigen-value have the same absolute eigen-value. Recently, the eigenvectors, especially the one associated with the largest eigenvalue, are also taking into
consider-* Corresponding author.
E-mail addresses:[email protected](G.-S. Lee),[email protected](C.-w. Weng). 0024-3795/$ – see front matter © 2013 Elsevier Inc. All rights reserved.
ation, for instances, in mathematical theory: [18,19,15,16,13,22]; in applications: [7,4]. See[6, pp. 65–69] for more applications. In this paper, we will give a (quasi-spectral) characterization of graphs when an identity involving eigenvalues, multiplicities, the eigenvector corresponding to the largest eigenvalue, and partial graph structure is satis-fied. The details are as follows.
Throughout this paper, let G be a connected graph with vertex set V , order n =|V |, diameter D, and distance function ∂. The adjacency matrix A of G is the binary matrix indexed by V , where the entry (A)uv = 1 if ∂(u, v) = 1, and (A)uv= 0 otherwise. Assume
that A has d+1 distinct eigenvalues λ0> λ1>· · · > λdwith corresponding multiplicities
m0= 1, m1, . . . , md. The spectrum of G is denoted by sp G ={λm00, λ
m1
1 , . . . , λ
md
d }, and
the parameter d is called the spectral diameter of G. Note that D d[3]. As is known, there is a sequence of orthogonal polynomials p0, p1, . . . , pd with respect to the inner
product , G (formally defined in the beginning of the next section), where deg pi = i
and pi, piG = pi(λ0) for 0 i d [15]. Let α be the eigenvector of A associated
with λ0 such that αtα = n and all entries of α are positive. Note that α is usually
called the Perron vector, and α = (1, 1, . . . , 1)t if and only if G is regular. For u ∈ V ,
let αu be the entry corresponding to u in the eigenvector α. For 0 i d, define the
weighted distance-i matrix Ai of G to be the matrix indexed by V such that the entry
( Ai)uv = αuαv if ∂(u, v) = i, and ( Ai)uv = 0 otherwise. In particular, for the case G
is regular, Ai is binary and is the so-called distance-i matrix Ai of G. For an integer
h d, we say that G is weighted h-punctually distance-regular if Ah = ph(A). Define
δi=
u,v( Ai◦ Ai)uv/n, where “◦” is the entrywise product of matrices. A bipartite graph
with bipartition (X, Y ) is called (k1, k2)-biregular if every vertex in X has degree k1and
every vertex in Y has degree k2. The distance-i graph of G is the graph whose adjacency
matrix is the distance-i matrix of G. For a connected bipartite graph G with bipartition (X, Y ), the halved graphs GXand GY are the two connected components of the distance-2
graph of G. It is well-known that the halved graphs of a bipartite distance-regular graph are distance-regular[5, Proposition 4.2.2].Examples 5.1–5.3show that the converse does not hold, that is, a connected bipartite graph whose halved graphs are distance-regular may not be distance-regular. Thus, a natural question is to find out when the converse is true. Our main result is the following.
Theorem 1.1. Let G be a connected bipartite graph with bipartition (X, Y ). Suppose that
G is weighted 2-punctually distance-regular with even spectral diameter, and both halved graphs GX and GY are distance-regular. Then G is distance-regular.
In addition to the main result, we believe that Proposition 3.3,Theorem 3.4, Propo-sition 4.5 andTheorem 5.6are of independent interest.
This paper is organized as follows. In the next section we provide some simple but useful lemmas for bipartite graphs. In Section 3, we present some results related to the spectral excess theorem [15], and characterize the graphs with δi = pi(λ0) for i ∈ {0, 1} (Lemma 3.7). In particular, this lemma is very useful for checking the
reg-ularity or biregreg-ularity of a graph. In Section 4, we study the concepts of weighted punctual distance-regularity and weighted partial distance-regularity, which can be re-garded as generalizations of the concepts of punctual distance-regularity and partial distance-regularity[11,10]. In Section 5, we proveTheorem 1.1.
2. Some results for bipartite graphs
In this section we provide some simple but useful lemmas to be used later on. These re-sults are related to the concept of orthogonal polynomials. The basic idea is to generalize the study of distance-regular graphs (see[3,5,24]).
2.1. Three-term recurrence
From the spectrum sp G ={λm0
0 , λ
m1
1 , . . . , λ
md
d } we consider the (d + 1)-dimensional
vector spaceRd[x] of real polynomials of degree at most d with inner product
p, qG:= d i=0 mi n p(λi)q(λi) = tr p(A)q(A)/n.
The predistance polynomials p0, p1, . . . , pd of G are orthogonal polynomials satisfying
deg pi= i andpi, piG = pi(λ0) for 0 i d [15]. Moreover, they satisfy a three-term
recurrence of the form
xpi= ci+1pi+1+ aipi+ bi−1pi−1 (1)
for 0 i d, where ci+1, ai, bi−1 are scalars in R, called the preintersection numbers
of G, with b−1 = cd+1 := 0 [12]. Note that ai+ bi+ ci = λ0 for 0 i d, where c0 := 0 and bd := 0 [8]. If G is bipartite, then ai = 0 for 0 i d [11], and thus
xpi= ci+1pi+1+ bi−1pi−1. By this observation, the predistance polynomials of bipartite
graphs satisfy a three-term recurrence of the form
x2pi= Xi+2pi+2+ Yipi+ Zi−2pi−2 (2)
for 0 i d, where Xi+2 := ci+1ci+2, Yi := bici+1+ bi−1ci and Zi−2 := bi−2bi−1. By
directly computing, it follows that Xi+ Yi+ Zi= λ20 for 0 i d. 2.2. The ‘odd’ or ‘even’ part
The sum of all predistance polynomials gives the Hoffman polynomial H [20]:
H(x) := n d i=1 x− λi λ0− λi = p0+ p1+· · · + pd, (3)
no matter whether the graph is regular or not. For a proof, see for instance[9]. Hoffman
[20] proved that a connected graph G is regular if and only if H(A) = J , the all-ones matrix. The following result, given first in[14, p. 117](see also[22, Lemma 2.1]) gives a generalization to nonregular graphs.
Lemma 2.1. Let G be a connected graph with adjacency matrix A and Perron vector α.
Then, H(A) = ααt. Moreover, G is regular if and only if H(A) = J , the all-ones matrix. 2
By the construction of Ai,Lemma 2.1and(3), any connected graph G has the property
that
A0+ A1+· · · + AD= H(A) = p0(A) + p1(A) +· · · + pd(A). (4)
If G is bipartite, then we can rewrite (4)(inLemma 2.2) more precisely by only taking the ‘odd’ or ‘even’ part, which was also considered in [11]. Define Aodd =
odd iAi,
podd=odd ipi and δodd=odd iδi. Similarly for Aeven, peven and δeven. For two n× n
real symmetric matrices M and N , define the inner product
M, N := 1 ntr(M N ) = 1 n u,v (M◦ N)uv.
Then A0, A1, . . . , AD are orthogonal, and δi = Ai, Ai. Hence A∗, A∗ = δ∗ and
p∗(A), p∗(A) = p∗(λ
0) for ∗ ∈ {odd, even}. If G is bipartite, then pi is odd or even
only depending on its degree i being odd or even [11]. The following lemma is proved by(4)and the fact that (pi(A))uv = 0 if ∂(u, v) and i have distinct parity (since bipartite
graphs contain no odd cycle).
Lemma 2.2. If G is bipartite, then Aodd= podd(A) and Aeven= peven(A). Moreover, by taking norms, δodd= podd(λ
0) and δeven = peven(λ0). 2
Remark 2.3. Observe that podd= (H(x)− H(−x))/2, H(λ0) = n and H(λd) = 0. Thus
for bipartite graphs, we deduce that δodd= δeven= podd(λ
0) = peven(λ0) = n/2.
3. The spectral excess theorem
The spectral excess theorem[15]asserts that δd pd(λ0) if G is regular, and equality
is attained if and only if G is distance-regular. See[9,17]for short proofs, and[11,10]for some generalizations. The parameter pd(λ0) is called the spectral excess of G, which can
be expressed in terms of the spectrum, which is
pd(λ0) = n π2 0 d i=0 1 miπi2 −1 ,
where πi =j=i|λi− λj| for 0 i d[15]. The following lemma gives an expression of
pd−1(λ0) for bipartite graphs in terms of the spectrum. The proof is essentially identical
to[9, p. 8–9], except the setting of the polynomials hi.
Lemma 3.1. Let G be a connected bipartite graph. Then
pd−1(λ0) = n 2 + d−1 i=1 (hi(λ0) + (−1)d−1hi(−λ0))2 mihi(λi)2 −1 , where hi= j=0,i,d(x− λj) for 1 i d − 1. 2 For 0 i d, define Ai =jiAj, pi = jipj and δi = jiδj. Similarly
for Ai, pi and δi. The parameter δD is referred to as the average weighted excess
and pD(λ0) as the generalized spectral excess of G. Recently, the authors [22] proved
the following ‘weighted’ version of the spectral excess theorem for nonregular graphs. In fact, the approach of giving weights, the entries of the Perron vector, to the vertices of a nonregular graph has been recently used many times in the literature (see, for instance,
[18,19,15,16,13]).
Theorem 3.2. (See [22].) Let G be a connected graph with diameter D. Then δD
pD(λ0) with equality if and only if AD = pD(A). Moreover, suppose further that
D = d. Then equality holds if and only if G is distance-regular. 2
Define Aodd
i = odd jiAj, δoddi = odd jiδj and poddi = odd jipj. Similarly
for A∗Ω, δΩ∗ and p∗Ω, where (Ω,∗) ∈ {(i, even), (i, odd), (i, even)}. Summing the recurrence relation(1) from the terms with index i + 1 to d, it follows that
xpi+1= bipi+ λ0pi+1− ci+1pi+1
[10, Proposition 2.5]. Note that if Ai+1= pi+1(A) and ∂(u, v) < i for u, v∈ V , then (Api+1(A))uv = 0 = (λ0pi+1(A))uv, and thus bi(pi(A))uv = ci+1(pi+1(A))uv. Using
this fact, we have the following result.
Proposition 3.3. Let G be a connected bipartite graph and i d − 1. Then Ai= pi(A)
if and only if Aj= pj(A) for 0 j i.
Proof. The sufficiency is clear. To prove necessity, we only need to show that Ai= pi(A)
(the remaining follows by similar argument). If ∂(u, v) i, then ( Ai)uv = (pi(A))uv by
assumption. Suppose ∂(u, v) < i. If ∂(u, v) and i have different parity, then ( Ai)uv =
0 = (pi(A))uv. Suppose that ∂(u, v) and i have the same parity. Then bi(pi(A))uv =
In [22], the authors posed the problem of characterizing the graphs which satisfy equality in Theorem 3.2 (or equivalently, AD = pD(A)), and gave a simple solution:
regular graphs with diameter 2 (in fact, these graphs are the so-called distance-polynomial
graphs[25]). Under the condition D = d, such graphs are distance-regular (Theorem 3.2). Here we complete this characterization for bipartite graphs.
Theorem 3.4. A connected bipartite graph with AD= pD(A) is distance-regular.
Proof. Note that the assumption is equivalent to AD−1 = pD−1(A). By Proposi-tion 3.3, Ai = pi(A) for 0 i D − 1. ByLemma 2.2, it follows that p∗D+1(A) is the
zero matrix, where ∗ ∈ {odd, even} has the same parity as D + 1. This happens only for the case D = d, since otherwise p∗D+1(λ0) = 0, contradicting the fact that pi(λ0) > 0
for 0 i d. The remaining follows fromTheorem 3.2. 2 Let
ProjNM := N, M N, NN
denote the projection of M onto Span{N}. Lemmas 3.5–3.6 present some inequalities related to the spectral excess theorem. The proofs are essentially the same as in [17, Lemma 1].
Lemma 3.5. Let G be a connected graph. For 0 i d,
(i) δi pi(λ0) with equality if and only if Ai= pi(A), and
(ii) δi pi(λ0) with equality if and only if Ai= pi(A). 2
Lemma 3.6. Let G be a connected bipartite graph. For0 i d and ∗ ∈ {odd, even}, (i) δi∗ p∗i(λ0) with equality if and only if A∗i= p∗i(A), and
(ii) δi∗ p∗i(λ0) with equality if and only if A∗i= p∗i(A). 2
A natural question motivated byLemmas 3.5–3.6is to study the relation between the parameters δiand pi(λ0) for 0 i d−1 (the case i = d is given inTheorem 3.2). We give
some results in the following. Note that p0= 1 and p1= λ0x/k (by the Gram–Schmidt
procedure), where k is the average degree of G. Moreover,
A1, A = 1 n u,v ( A1)uv= 1 n1 tA 11 = 1 n1 tDAD1 = λ 0, (5)
where 1 is the all-ones vector, and D is the diagonal matrix with entries Duu = αu
the regularity of a graph, which follows from the inequality δ0 p0(λ0) mentioned in
Lemma 3.5. In fact, it can also be derived by the Cauchy–Schwarz inequality:u∈V α4u
(u∈V α2
u)2/n = n. The second part ofLemma 3.7 characterizes the graphs satisfying
δ1= p1(λ0), which is useful for checking the regularity or biregularity of a graph.
Lemma 3.7. Let G be a connected graph. Then
1. δ0 1 (= p0(λ0)) (which is equivalent tou∈V α4u n), with equality if and only if
any of the following conditions holds:
(i) A0= I (= p0(A)),
(ii) G is regular.
2. δ1 λ20/k (= p1(λ0)), with equality if and only if any of the following conditions holds:
(i) A1= p1(A),
(ii) G is regular or biregular.
Proof. We only need to prove the second part. Computing ProjA
1p1(A) by the same
argument as in[17, Lemma 1]and(5), it follows that δ1 p1(λ0), with equality if and
only if A1= p1(A). Now it remains to show that (i)⇔ (ii). To prove necessity, we give
the weight αuto the vertex u∈ V , and the weight αuαv to the edge connecting u and v.
Since A1= p1(A) = λ0A/k, all edges receive the same weight, λ0/k. If G is not bipartite,
then it contains an odd cycle, and all vertices on this cycle must have the same weight. The assumption ‘G is connected’ deduces that all vertices are of the same weight. Thus
G is regular. For the case G is bipartite, the condition ‘all edges receive the same weight’
implies that vertices in the same partite set have the same weight. Thus G is biregular. Now we prove sufficiency. If G is regular, then clearly p1(A) = λ0A/k = A = A1. Suppose
that G is (k1, k2)-biregular with bipartition (X, Y ), where|X| = n1,|Y | = n2. Note that λ0=
√
k1k2, n1k1= n2k2 and the Perron vector α =α, . . . , α n1 α, . . . , α n2 t , where α=n1+n2 2n1 and α =n1+n2 2n2 . Thus p1(A) = λ0 k A = √ k1k2(n1+ n2) n1k1+ n2k2 A = n1+ n2 2√n1n2 A = ααA = A1. 2
The next question is to discuss the relation between δ2 and p2(λ0). We give the
answer under the assumption G is regular, and provide an example to show that the regularity condition is necessary. Thus, there is no hope to determine the order of δ2and p2(λ0) uniformly. Lemma 3.8is proved by the inequality δ2 p2(λ0) mentioned in
Lemma 3.8. Let G be a connected regular graph. Then δ2 p2(λ0), with equality if and only if A2= p2(A). 2
Example 3.9. (See[22].) Let P3be a path of three vertices, with spectrum{ √
2, 0,−√2}. Then D = d = 2, p0= 1, p1= 3
√
2x/4, p2= 3(x2−4/3)/4, and δ2= 3/8 < 1/2 = p2(λ0).
4. Weighted punctual and partial distance-regularity
The concepts of punctual distance-regularity and partial distance-regularity have been recently studied [11,10]. In this paper, we study two concepts, which are basically the same as that in [11,10], except that here we drop the regularity assumption, and the use of weighted distance matrices is taking into account. A connected graph is called
weighted h-punctually distance-regular if Ah= ph(A); and is called weighted m-partially
distance-regular if Ai = pi(A) for i m. Note by Lemma 3.7 that the regularity
con-dition is actually not necessary in the concept of partial distance-regularity. Clearly, the concepts of weighted 0-punctual regularity and weighted 0-partial distance-regularity are identical. However, the weighted 1-punctual distance-distance-regularity and the weighted 1-partial distance-regularity are not equivalent. For example, by Lemma 3.7, the path graph P3 of three vertices is weighted 1-punctually distance-regular, but not
weighted 1-partially distance-regular. The following result indicates that the concepts of weighted 2-punctual distance-regularity and weighted 2-partial distance-regularity coin-cide.
Proposition 4.1. Let G be a connected graph. Then A2 = p2(A) if and only if G is weighted 2-partially distance-regular.
Proof. We only need to prove necessity. Since A2= p2(A) = aA2+ bA + cI for some real
numbers a, b, c with a= 0, we conclude that A2has a constant diagonal, which implies
that G is regular. The remaining follows from Lemma 3.7. 2
Proposition 4.2 states an equivalent condition of the weighted 2-punctual distance-regularity for bipartite graphs with spectral diameter d 3. Note that the assumption
d 3 is necessary, since otherwise the path graph P3of three vertices gives a
counterex-ample. The proof follows from Proposition 3.3,Lemma 3.5and Proposition 4.1.
Proposition 4.2. Let G be a connected bipartite graph with spectral diameter d 3. Then δ2= p2(λ0) if and only if G is weighted 2-punctually distance-regular. 2
Lemma 4.3 demonstrates that for a connected bipartite weighted 2-punctually distance-regular graph, its two halved graphs have the same spectrum (with appropriate spectral diameter), and, under further assumption, it gives a lower bound or exact value of the diameter, depending on the parity of its spectral diameter.
Lemma 4.3. Let G be a connected bipartite graph with bipartition (X, Y ), diameter D,
spectral diameter d and A2= p2(A). Then the halved graphs GX and GY have the same spectrum, and are of spectral diameter d/2 . Suppose further that at least one of GX
and GY has spectral diameter which is equal to its diameter. Then D d − 1 for odd d,
and D = d otherwise.
Proof. Since G is bipartite, p2 is even, that is, p2 = ax2+ b for some real numbers a, b
with a= 0. Let X1and Y1be adjacency matrices of GX and GY, respectively. Note that
A =
0 B BT 0
for some square matrix B (since G is regular byProposition 4.1). Hence X1 0 0 Y1 = A2= p2(A) = aA2+ bI = aBBT + bI 0 0 aBTB + bI .
Since BBT and BTB have the same characteristic polynomial (see for instance [26, Theorem 2.8]), GX and GY have the same spectrum. Note that if λ is an eigenvalue
of A with eigenvector u then aλ2+ b is an eigenvalue of A
2 with the same eigenvector.
Thus A2has (d + 1)/2 = d/2 + 1 distinct eigenvalues, and so do GX and GY. Hence GX and GY are of spectral diameterd/2 . If at least one of GX and GY has spectral
diameter which is equal to its diameter, we derived that d D 2d/2 , as claimed. 2 At the end of this section, we give some results for connected bipartite graphs with δd−1 = pd−1(λ0).Lemma 4.4 follows fromLemma 3.6. The proof ofProposition 4.5 is
basically identical to[17, Proposition 2] (it is not difficult to prove this characterization by backward induction on i, using the recurrence relation(2),Lemmas 2.2, 4.4 and 3.7). Lemma 4.4. Let G be a connected bipartite graph. Then δd−1 pd−1(λ0), with equality if and only if Ad−1= pd−1(A). 2
Proposition 4.5. Let G be a connected bipartite graph with δd−1= pd−1(λ0). Then Ai=
pi(A) for all i with the opposite parity of d. In particular, G is regular if d is odd, and
biregular otherwise. 2
5. Proof of the main result
It is well-known that the halved graphs of a bipartite distance-regular graph are distance-regular [5, Proposition 4.2.2]. We first provide three examples to show that the converse does not hold, that is, a connected bipartite graph whose halved graphs are distance-regular may not be distance-regular. Here we omit the computation details which are straightforward by definitions.
Example 5.1 (Weighted 2-punctually distance-regular and odd spectral diameter). Consider the Möbius–Kantor graph, i.e., the generalized Petersen graph G(8, 3) [23], with spectrum{31,√34, 13, (−1)3, (−√3)4, (−3)1}. Then D = 4 < 5 = d, p0= 1, p1= x, p2= x2− 3, p3= 2(x3− 5x)/5, p4= (x4− 10x2+ 15)/6, p5= (x5− 56x3/5 + 21x)/18,
Ai= pi(A) for i∈ {0, 1, 2, 4}, and both halved graphs are distance-regular with spectrum
{61, 04, (−2)3}.
Example 5.2 (Not weighted 2-punctually distance-regular and even spectral diameter). Consider the Hoffman graph with spectrum{41, 24, 06, (−2)4, (−4)1}, which is cospectral
to the Hamming 4-cube but not distance-regular[20,5]. Then D = d = 4, p0= 1, p1= x, p2= (x2−4)/2, p3= (x3−10x)/6, p4= (x4−16x2+ 24)/24, Ai= pi(A) for i∈ {0, 1, 3},
and its two halved graphs are the complete graph K8and the complete multipartite graph K2,2,2,2, which are both distance-regular.
Example 5.3 (Not weighted 2-punctually distance-regular and odd spectral diame-ter). Consider the graph obtained by deleting a 10-cycle from the complete bipartite
graph K5,5, with spectrum {31, (( √ 5 + 1)/2)2, ((√5− 1)/2)2, ((−√5 + 1)/2)2, ((−√5− 1)/2)2, (−3)1}. Then D = 3 < 5 = d, p 0 = 1, p1 = x, p2 = 3(x2 − 3)/5, p3 = 12(x3− 19x/3)/49, p 4= (x4− 48x2/5 + 49/5)/11, p5= (x5− 543x3/49 + 2820x/147)/33,
Ai= pi(A) for i∈ {0, 1}, and both halves graphs are the complete graphs K5, which are
distance-regular.
The following result is related to [5, Proposition 4.2.2](in the case that d is even). Theorem 5.4. Let G be a connected bipartite graph with bipartiton (X, Y ) and spectral
diameter d. Suppose that Ai = pi(A) for even i, where 0 i d. Then G is weighted
2-punctually distance-regular and both halved graphs GX and GY are distance-regular with diameter d/2 .
Proof. By assumption, A0 = p0(A) = I and A2 = p2(A) = aA2+ bI for some real
numbers a, b with a= 0. Then G is regular and weighted 2-punctually distance-regular. ByLemma 4.3, GXand GY have the same spectrum, and are of spectral diameterd/2 . Since p2i is even, we can assume p2i= fi(ax2+ b) for some fi ∈ R[x] of degree i. Thus,
for 0 i d/2 , Xi 0 0 Yi = A2i= p2i(A) = fi aA2+ bI= fi( A2) = fi(X1) 0 0 fi(Y1) ,
where Xiand Yi are distance-i matrices of GX and GY, respectively. Therefore, GX and
GY are distance-regular with diameterd/2 . 2
Theorem 5.5. Let G be a connected bipartite graph with bipartition (X, Y ) and spectral
di-ameter d. Suppose that G is weighted 2-punctually distance-regular and both halved graphs GX and GY are distance-regular with diameterd/2 . Then δ
= p(λ0), where = d−1 if d is odd, and = d otherwise. In particular, if d is even, then Theorem 1.1holds.
Proof. First note byProposition 4.1that G is regular. ByLemma 4.3, GX and GY have the same spectrum, and are of spectral diameterd/2 . Thus GX and GY have the same
(pre)distance-polynomials fi, 0 i d/2 . Since GX and GY are distance-regular,
A2i= Xi 0 0 Yi = fi(X1) 0 0 fi(Y1) = fi( A2) = fi p2(A) = g2i(A)
for 0 i d/2 , where Xi and Yi are distance-i matrices of GX and GY, respectively,
and g2i∈ R[x] is even of degree 2i. Since G is regular, AJ = g(A)J = g(λ0)J . Then
each row of A has exactly g(λ0) ones, and thus δ = g(λ0). Now it remains to show
that g = p. Note that g, gG =g(A), g(A) = A, A = δ = g(λ0). For every
polynomial p∈ R−1[x],g, pG= A, p(A) = 0. By the uniqueness of the predistance
polynomials, it follows that g= p. Moreover, if d is even, then byTheorem 3.2, G is
distance-regular. 2
Putting Proposition 4.5, Theorems 3.2, 5.4 and 5.5 together, we can conclude the following theorem.
Theorem 5.6. Let G be a connected bipartite graph with bipartition (X, Y ) and spectral
diameter d. Then the following conditions are equivalent.
(i) Ai= pi(A) for even i, where 0 i d;
(ii) δ= p(λ0), where = d− 1 if d is odd, and = d otherwise;
(iii) G is weighted 2-punctually distance-regular and both halved graphs GX and GY are
distance-regular with diameterd/2 . 2
Applying a result in[1, Theorem 4.2], Theorem 5.6(i) seems to be improved to the condition that only i∈ {0, d − 2} is necessary when d is even. Unfortunately, there is a flaw in the proof of this result[2].
Note that the Möbius–Kantor graph (Example 5.1) with odd spectral diameter satis-fiesTheorem 5.6(i)–(iii) with D = d− 1. The following example shows that a bipartite graph with odd spectral diameter satisfying Theorem 5.6(i)–(iii) and D = d needs not to be distance-regular.
Example 5.7. Consider the regular bipartite graphs on 20 vertices obtained from the Desargues graph by the Godsil–McKay switching, which is not distance-regular with spectrum{31, 24, 15, (−1)5, (−2)4, (−3)1} [21]. Then D = d = 5, p0 = 1, p1 = x, p2 = x2− 3, p3 = (x3− 5x)/2, p4 = (x4− 9x2+ 12)/4, p5 = (x5− 11x3+ 22x)/12, Ai =
pi(A) for i ∈ {0, 1, 2, 4}, and both halved graphs are distance-regular with spectrum
{61, 14, (−2)5}.
Acknowledgements
The authors are grateful to the anonymous referees for careful reading and valuable suggestions that improved the presentation. This research is supported by the National Science Council of Taiwan ROC under the project NSC 102-2115-M-009-MY3.
References
[1]A. Abiad, C. Dalfó, M.A. Fiol, Algebraic characterizations of regularity properties in bipartite graphs, European J. Combin. 34 (2013) 1223–1231.
[2] A. Abiad, C. Dalfó, M.A. Fiol, Corrigendum to “Algebraic characterizations of regularity properties in bipartite graphs” [European J. Combin. 34 (2013) 1223–1231], European J. Combin. (2013),
http://dx.doi.org/10.1016/j.ejc.2013.11.003.
[3]N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, 1993.
[4]D. Bu, Y. Zhao, L. Cai, H. Xue, X. Zhu, H. Lu, J. Zhang, S. Sun, L. Ling, N. Zhang, G. Li, R. Chen, Topological structure analysis of the protein–protein interaction network in budding yeast, Nucleic Acids Res. 31 (2003) 2443–2450.
[5]A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989.
[6]A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer, 2012.
[7]S. Brin, L. Page, The anatomy of a large-scale hypertextual Web search engine, Comput. Netw. ISDN Syst. 30 (1998) 107–117.
[8]M. Cámara, J. Fàbrega, M.A. Fiol, E. Garriga, Some families of orthogonal polynomials of a discrete variable and their applications to graphs and codes, Electron. J. Combin. 16 (1) (2009), #R83.
[9]E.R. van Dam, The spectral excess theorem for distance-regular graphs: a global (over)view, Elec-tron. J. Combin. 15 (1) (2008), #R129.
[10]C. Dalfó, E.R. van Dam, M.A. Fiol, E. Garriga, Dual concepts of almost distance-regularity and the spectral excess theorem, Discrete Math. 312 (2012) 2730–2734.
[11]C. Dalfó, E.R. van Dam, M.A. Fiol, E. Garriga, B.L. Gorissen, On almost distance-regular graphs, J. Combin. Theory Ser. A 118 (2011) 1094–1113.
[12]C. Dalfó, M.A. Fiol, E. Garriga, Characterizing (, m)-walk-regular graphs, Linear Algebra Appl. 433 (2010) 1821–1826.
[13]M.A. Fiol, Eigenvalue interlacing and weight parameters of graphs, Linear Algebra Appl. 290 (1999) 275–301.
[14]M.A. Fiol, Algebraic characterizations of distance-regular graphs, Discrete Math. 246 (2002) 111–129.
[15]M.A. Fiol, E. Garriga, From local adjacency polynomials to local pseudo-distance-regular graphs, J. Combin. Theory Ser. B 71 (1997) 162–183.
[16]M.A. Fiol, E. Garriga, On the algebraic theory of pseudo-distance-regularity around a set, Linear Algebra Appl. 298 (1999) 115–141.
[17]M.A. Fiol, S. Gago, E. Garriga, A simple proof of the spectral excess theorem for distance-regular graphs, Linear Algebra Appl. 432 (2010) 2418–2422.
[18]M.A. Fiol, E. Garriga, J.L.A. Yebra, On a class of polynomials and its relation with the spectra and diameters of graphs, J. Combin. Theory Ser. B 67 (1996) 48–61.
[19]M.A. Fiol, E. Garriga, J.L.A. Yebra, Locally pseudo-distance-regular graphs, J. Combin. Theory Ser. B 68 (1996) 179–205.
[20]A.J. Hoffman, On the polynomial of a graph, Amer. Math. Monthly 70 (1963) 30–36.
[21]W.H. Haemers, E. Spence, Graphs cospectral with distance-regular graphs, Linear Multilinear Al-gebra 39 (1995) 91–107.
[22]G.-S. Lee, C.-w. Weng, A spectral excess theorem for nonregular graphs, J. Combin. Theory Ser. A 119 (2012) 1427–1431.
[23]D. Marušič, T. Pisanski, The remarkable generalized Petersen graph G(8, 3), Math. Slovaca 50 (2000) 117–121.
[24]P. Terwilliger, The subconstituent algebra of an association scheme I, J. Algebraic Combin. 1 (1992) 363–388.
[25]P.M. Weichsel, On distance-regularity in graphs, J. Combin. Theory Ser. B 32 (1982) 156–161.
