Journal of Algebraic Combinatorics 10 (1999), 47–59
c 1999 Kluwer Academic Publishers. Manufactured in The Netherlands.
°
Tight Graphs and Their Primitive Idempotents
ARLENE A. PASCASIO
Mathematics Department, De La Salle University, Manila 1004, Philippines
Abstract.
In this paper, we prove the following two theorems.
Theorem 1 Let 0 denote a distance-regular graph with diameter d ≥ 3. Suppose E and F are primitive
idempotents of 0, with cosine sequences σ0 , σ1 , . . . , σd and ρ0 , ρ1 , . . . , ρd , respectively. Then the following are
equivalent.
(i) The entry-wise product E ◦ F is a scalar multiple of a primitive idempotent of 0.
(ii) There exists a real number ² such that
σi ρi − σi−1 ρi−1 = ²(σi−1 ρi − σi ρi−1 )
(1 ≤ i ≤ d).
Let 0 denote a distance-regular graph with diameter d ≥ 3 and eigenvalues θ0 > θ1 > · · · > θd . Then Juris̆ić,
Koolen and Terwilliger proved that the valency k and the intersection numbers a1 , b1 satisfy
µ
θ1 +
k
a1 + 1
¶µ
θd +
k
a1 + 1
¶
≥
−ka1 b1
.
(a1 + 1)2
They defined 0 to be tight whenever 0 is not bipartite, and equality holds above.
Theorem 2 Let 0 denote a distance-regular graph with diameter d ≥ 3 and eigenvalues θ0 > θ1 > · · · > θd .
Let E and F denote nontrivial primitive idempotents of 0.
(i) Suppose 0 is tight. Then E, F satisfy (i), (ii) in Theorem 1 if and only if E, F are a permutation of E 1 , E d .
(ii) Suppose 0 is bipartite. Then E, F satisfy (i), (ii) in Theorem 1 if and only if at least one of E, F is equal to
Ed .
(iii) Suppose 0 is neither bipartite nor tight. Then E, F never satisfy (i), (ii) in Theorem 1.
Keywords:
1.
tight graph, distance-regular, association scheme, Krein parameter
Introduction
Let 0 denote a distance-regular graph with vertex set X and diameter d ≥ 3. Let E 0 ,
E 1 , . . . , E d denote the primitive idempotents of 0 (see definitions in the next section).
It is well-known
E i ◦ E j = |X |−1
d
X
qihj E h
(0 ≤ i, j ≤ d),
(1)
h=0
This work was done when the author was an Honorary Fellow at the University of Wisconsin-Madison (September
1996–August 1997) supported by the Department of Science and Technology, Philippines.
48
PASCASIO
where ◦ denotes the entry-wise matrix product and where qihj are the Krein parameters.
By (1), and since E 0 , E 1 , . . . , E d are linearly independent we see that for all integers
i, j (0 ≤ i, j ≤ d), the following are equivalent.
(i) E i ◦ E j is a scalar multiple of a primitive idempotent of 0.
6 0 for exactly one h ∈ {0, 1, . . . , d}.
(ii) qihj =
In this paper, we investigate pairs E i and E j for which (i), (ii) hold above. We state our
main results in Theorems 1.1 and 1.3 below.
Theorem 1.1 Let 0 denote a distance-regular graph with diameter d ≥ 3. Suppose E and
F are primitive idempotents of 0, with cosine sequences σ0 , σ1 , . . . , σd and ρ0 , ρ1 , . . . , ρd ,
respectively. Then the following are equivalent.
(i) E ◦ F is a scalar multiple of a primitive idempotent of 0.
(ii) There exists a real number ² such that
σi ρi − σi−1 ρi−1 = ²(σi−1 ρi − σi ρi−1 ) (1 ≤ i ≤ d).
Let 0 denote a distance-regular graph with diameter d ≥ 3 and eigenvalues θ0 > θ1 > · · · >
θd . In [4], Juris̆ić, Koolen and Terwilliger proved that the valency k and the intersection
numbers a1 , b1 satisfy
¶µ
¶
µ
−ka1 b1
k
k
θd +
≥
.
(2)
θ1 +
a1 + 1
a1 + 1
(a1 + 1)2
They defined 0 to be tight whenever 0 is not bipartite and equality holds in (2). They showed
that 0 is tight precisely when 0 is “1-homogeneous” and ad = 0. They also obtained the
following characterization of tight graphs, which will be useful later.
Theorem 1.2 ([4]) Let 0 denote a nonbipartite distance-regular graph with diameter
d ≥ 3, and eigenvalues θ0 > θ1 > · · · > θd . Let θ and θ 0 denote eigenvalues of 0,
with respective cosine sequences σ0 , σ1 , . . . , σd and ρ0 , ρ1 , . . . , ρd . Let ² ∈ R. Then the
following are equivalent.
σρ − 1
.
(i) 0 is tight, θ, θ 0 are a permutation of θ1 , θd and ² =
ρ−σ
0
(ii) θ 6= θ0 , θ 6= θ0 , and
σi ρi − σi−1 ρi−1 = ²(σi−1 ρi − σi ρi−1 ) (1 ≤ i ≤ d).
Combining Theorems 1.1 and 1.2, we obtain the following result.
Theorem 1.3 Let 0 denote a distance-regular graph with diameter d ≥ 3, and eigenvalues
θ0 > θ1 > · · · > θd . Let E and F denote nontrivial primitive idempotents of 0.
(i) Suppose 0 is tight. Then E, F satisfy (i), (ii) in Theorem 1.1 if and only if E, F are a
permutation of E 1 , E d .
(ii) Suppose 0 is bipartite. Then E, F satisfy (i), (ii) in Theorem 1.1 if and only if at least
one of E, F is equal to E d .
(iii) Suppose 0 is neither bipartite nor tight. Then E, F never satisfy (i), (ii) in Theorem 1.1.
TIGHT GRAPHS AND THEIR PRIMITIVE IDEMPOTENTS
2.
49
Preliminaries
In this section, we review some definitions and basic concepts. For more background information, the reader may refer to the books of Bannai and Ito [1], Brouwer et al. [2] or
Godsil [3].
Throughout, 0 will denote a finite, undirected, connected graph without loops or multiple
edges, with vertex set X , path-length distance function ∂ and diameter d := max{∂(x, y) |
x, y ∈ X }. We say 0 is distance-regular whenever for all integers h, i, j (0 ≤ h, i, j ≤ d)
and for all x, y ∈ X with ∂(x, y) = h, the number
pihj := |{z ∈ X | ∂(x, z) = i, ∂(y, z) = j}|
is independent of x and y. The integers pihj are called the intersection numbers for 0. We
i
i
i
(0 ≤ i ≤ d), bi := p1i+1
(0 ≤ i ≤ d − 1), ci := p1i−1
(1 ≤ i ≤ d),
abbreviate ai := p1i
0
and ki := pii (0 ≤ i ≤ d). Observe
ci + ai + bi = k
(0 ≤ i ≤ d),
(3)
where k := k1 = b0 , c0 := 0 and bd := 0.
It is known, by [1], (Chapter 3, Proposition 1.2)
ki =
b0 b1 b2 · · · bi−1
c1 c2 c3 · · · ci
(0 ≤ i ≤ d).
(4)
Hereon, we assume 0 is distance-regular.
Let Mat X (C) denote the algebra of matrices over C with rows and columns indexed by
X . For each integer i (0 ≤ i ≤ d), the ith distance matrix Ai ∈ Mat X (C) has x, y entry
(Ai )x y =
½
1,
if ∂(x, y) = i
0,
if ∂(x, y) 6= i
(x, y ∈ X ).
Then
A0 = I,
A0 + A1 + · · · + Ad = J,
Ait = Ai
Ai A j =
d
X
(0 ≤ i ≤ d),
pihj Ah
(0 ≤ i, j ≤ d),
h=0
where J denotes the all 1’s matrix. The matrices A0 , A1 , . . . , Ad form a basis for a commutative semi-simple C-algebra M, called the Bose-Mesner algebra of 0. By [1], (Section 2.3)
50
PASCASIO
M has a second basis E 0 , E 1 , . . . , E d such that
E 0 = |X |−1 J,
E 0 + E 1 + · · · + E d = I,
E it = E i
(5)
(0 ≤ i ≤ d),
E i E j = δi j E i
(0 ≤ i, j ≤ d).
The E 0 , E 1 , . . . , E d are called the primitive idempotents of 0, and E 0 is called the trivial
idempotent.
Set A := A1 and define θ0 , θ1 , . . . , θd ∈ C such that
A=
d
X
θi E i .
i=0
It is known θ0 = k, and that θ0 , θ1 , . . . , θd are distinct real numbers. Moreover, k ≥ θi ≥
−k (0 ≤ i ≤ d), see [1], (Theorem 1.3). We refer to θi as the eigenvalue of 0 associated
with E i (0 ≤ i ≤ d). We call θ0 the trivial eigenvalue of 0. For each integer i(0 ≤ i ≤ d),
let m i denote the rank of E i . We refer to m i as the multiplicity of E i (or θi ). We observe
from (5) that m 0 = 1.
Let θ denote an eigenvalue of 0, let E denote the associated primitive idempotent, and
let m denote the multiplicity of E. By [1], (Section 2.3) there exist σ0 , σ1 , . . . , σd ∈ R such
that
E = |X |−1 m
d
X
σi Ai .
i=0
It follows from [1], (Chapter 2, Proposition 3.3 (iii)) that σ0 = 1. We call σ0 , σ1 , . . . , σd
the cosine sequence of 0 associated with E (or θ ). The cosine sequence associated with
E 0 consists entirely of ones, see [2], (Section 4.1B). We shall often denote σ1 by σ .
Lemma 2.1 Let 0 denote a distance-regular graph with diameter d ≥ 3. For any θ, σ0 ,
σ1 , . . . , σd ∈ C, the following are equivalent.
(i) σ0 , σ1 , . . . , σd is a cosine sequence of 0 and θ is the associated eigenvalue.
(ii) σ0 = 1, and
ci σi−1 + ai σi + bi σi+1 = θσi
(0 ≤ i ≤ d),
(6)
where σ−1 and σd+1 are indeterminates.
(iii) σ0 = 1, kσ = θ and
ci (σi−1 − σi ) − bi (σi − σi+1 ) = k(σ − 1)σi
where σd+1 is an indeterminate.
(1 ≤ i ≤ d),
(7)
TIGHT GRAPHS AND THEIR PRIMITIVE IDEMPOTENTS
51
Proof:
(i) ⇔ (ii) See [2], (Proposition 4.1.1).
(ii) ⇔ (iii) Follows from (3).
2
Lemma 2.2 (Christoffel-Darboux Formula) Let 0 denote a distance-regular graph with
diameter d ≥ 3. For cosine sequences σ0 , σ1 , . . . , σd and ρ0 , ρ1 , . . . , ρd of 0,
(σ − ρ)
i
X
k h σh ρh =
h=0
b1 b2 · · · bi
(σi+1 ρi − σi ρi+1 ) (0 ≤ i ≤ d),
c1 c2 · · · ci
(8)
where σd+1 and ρd+1 are indeterminates.
Proof:
See [1], (Theorem 1.3) or [3], (Lemma 3.1).
2
Corollary 2.3 Let 0 denote a distance-regular graph with diameter d ≥ 3, and let σ0 ,
σ1 , . . . , σd denote a sequence of complex numbers. Then the following are equivalent.
(i) σ0 = 1, and
(σ − 1)
i
X
k h σh =
h=0
b1 b2 · · · bi
(σi+1 − σi ) (0 ≤ i ≤ d),
c1 c2 · · · ci
(9)
where σd+1 is an indeterminate.
(ii) σ0 , σ1 , . . . , σd is a cosine sequence of 0.
Proof:
(i) ⇒ (ii) We show (7) holds. Pick an integer i(1 ≤ i ≤ d). By (9) (with i replaced by i − 1)
we have
(σ − 1)
i−1
X
h=0
k h σh =
b1 b2 · · · bi−1
(σi − σi−1 ).
c1 c2 · · · ci−1
(10)
Subtracting Eq. (10) from Eq. (9) and eliminating ki from the result using (4), we get (7)
as desired. Now σ0 , σ1 , . . . , σd is a cosine sequence by Lemma 2.1 (i) and (iii).
(ii) ⇒ (i) Observe σ0 = 1 by Lemma 2.1 (ii). To obtain (9), in Lemma 2.2 let ρ0 , ρ1 , . . . , ρd
denote the cosine sequence for E 0 , that is, ρi = 1 (0 ≤ i ≤ d).
2
The graph 0 is said to be bipartite whenever ai = 0 for 0 ≤ i ≤ d.
Lemma 2.4 Let 0 denote a distance-regular graph with diameter d ≥ 3, valency k, and
eigenvalues θ0 > θ1 > · · · > θd . Let σ0 , σ1 , . . . , σd denote the cosine sequence for θd .
Then the following are equivalent.
52
PASCASIO
(i) 0 is bipartite.
(ii) θd = −k.
(iii) σi = (−1)i (0 ≤ i ≤ d).
Proof:
(i) ⇔ (ii) See [2], (Proposition 3.2.3).
(i),(ii) ⇒ (iii) Follows easily from Lemma 2.1 (i), (ii).
(iii) ⇒ (ii) Observe σ1 = −1, and θd = σ1 k by Lemma 2.1 (i), (iii), so θd = −k.
2
Lemma 2.5 Let 0 denote a bipartite distance-regular graph with diameter d ≥ 3, and
eigenvalues θ0 > θ1 > · · · > θd . Pick any integer i (0 ≤ i ≤ d). Then
(i) θi = −θd−i ,
(ii) m i = m d−i ,
(iii) Let σ0 , σ1 , . . . , σd denote the cosine sequence for θi . Then the cosine sequence for
θd−i is σ0 , −σ1 , σ2 , −σ3 , . . . , (−1)d σd .
Proof:
(i), (ii) See [2], (Proposition 3.2.3).
(iii) By Lemma 2.1 (ii) and recalling that a0 , a1 , . . . , ad are all 0, it suffices to show
c j (−1) j−1 σ j−1 + b j (−1) j+1 σ j+1 = θd−i (−1) j σ j
(0 ≤ j ≤ d).
(11)
By Lemma 2.1 (i), (ii), and since σ0 , σ1 , . . . , σd is a cosine sequence for θi ,
c j σ j−1 + b j σ j+1 = θi σ j
(0 ≤ j ≤ d).
Evaluating (12) using Lemma 2.5 (i) we obtain (11), as desired.
(12)
2
Let σ0 , σ1 , . . . , σd denote a finite sequence of nonzero real numbers. By the number of
sign changes in this sequence we mean the number of integers i (0 ≤ i ≤ d − 1) such that
σi σi+1 < 0. For an arbitrary finite sequence of real numbers, the number of sign changes
in it is the number of sign changes in the sequence obtained by disregarding the zero
terms.
Lemma 2.6 Let 0 denote a distance-regular graph with diameter d ≥ 3, and eigenvalues
θ0 > θ1 > · · · > θd . Let σ0 , σ1 , . . . , σd denote a cosine sequence of 0, and let θ denote the
corresponding eigenvalue. For any integer i(0 ≤ i ≤ d), the following are equivalent.
(i) θ = θi .
(ii) σ0 , σ1 , . . . , σd has exactly i sign changes.
Moreover, suppose i ≥ 1, and that (i), (ii) hold. Then the sequence σ0 − σ1 , σ1 − σ2 , . . . ,
σd−1 − σd has exactly i − 1 sign changes.
TIGHT GRAPHS AND THEIR PRIMITIVE IDEMPOTENTS
53
Proof:
2
See [2], (Corollary 4.1.2) or [3], (Lemma 2.1).
Lemma 2.7 Let 0 denote a distance-regular graph with diameter d ≥ 3, and eigenvalues
θ0 > θ1 > · · · > θd . Let σ0 , σ1 , . . . , σd denote the cosine sequence associated with θ1 .
Then
σ0 > σ1 > · · · > σd .
(13)
Proof: By Lemma 2.6, the sequence σ0 − σ1 , σ1 − σ2 , . . . , σd−1 − σd has no sign changes.
Recall, σ1 = θ1 k −1 and σ0 = 1 by Lemma 2.1 (iii), so
1 = σ0 > σ1 ≥ σ2 ≥ · · · ≥ σd .
(14)
Suppose (13) fails. Then there exists an integer i (1 ≤ i ≤ d − 1) such that
σi−1 > σi = σi+1 .
(15)
Setting σi = σi+1 in (7), we find in view of (15) that
σi = σi+1 < 0.
(16)
Assume for now that i = d − 1, so σd−1 = σd < 0. Using (15), and by setting i = d in (7)
we obtain
0 = cd (σd−1 − σd )
= k(σ − 1)σd ,
so σd = 0, a contradiction. Hence, i < d − 1. We may now argue, by (14), (15), (7) and
(16)
0 ≥ −bi+1 (σi+1 − σi+2 )
= −bi+1 (σi+1 − σi+2 ) + ci+1 (σi − σi+1 )
= k(σ − 1)σi+1
> 0,
a contradiction. We now have (13), as desired.
2
Let 0 denote a distance-regular graph with diameter d ≥ 3, and let M denote the Bose
Mesner algebra of 0. Since M is closed under the entrywise matrix product ◦, there exist
scalars qihj ∈ C such that
E i ◦ E j = |X |−1
d
X
qihj E h
(0 ≤ i, j ≤ d).
h=0
We call the qihj the Krein parameters of 0.
(17)
54
3.
PASCASIO
The main results
Let E and F denote primitive idempotents of a distance-regular graph 0. In this section,
we focus on finding necessary and sufficient conditions such that E ◦ F is a scalar multiple
of a primitive idempotent of 0.
Lemma 3.1 Let 0 denote a distance-regular graph with diameter d ≥ 3. For all integers
h, i, j (0 ≤ h, i, j ≤ d), the following are equivalent.
(i) E i ◦ E j is a scalar multiple of E h .
(ii) qirj = 0 for all r ∈ {0, 1, . . . , d}\h.
Proof:
Follows immediately from (17) and the linear independence of E 0 , E 1 , . . . , E d .
2
Lemma 3.2 Let 0 denote a distance-regular graph with diameter d ≥ 3. Let E, F and H
denote primitive idempotents of 0, and let σ0 , σ1 , . . . , σd ; ρ0 , ρ1 , . . . , ρd and γ0 , γ1 , . . . , γd
denote the associated cosine sequences. Then the following are equivalent.
(i) E ◦ F is a scalar multiple of H .
(ii) γi = σi ρi (0 ≤ i ≤ d).
Moreover, suppose (i), (ii) hold. Then the scalar referred to in (i) is equal to
−1
m σ m ρ m −1
γ |X | ,
(18)
where m σ , m ρ and m γ denote the multiplicity of E, F and H respectively.
Proof:
Observe
E = |X |−1 m σ
d
X
σi Ai ,
(19)
ρi Ai ,
(20)
γi Ai .
(21)
i=0
F = |X |−1 m ρ
d
X
i=0
H = |X |−1 m γ
d
X
i=0
(i) ⇒ (ii) By assumption, there exists α ∈ C such that
E ◦ F = α H.
(22)
Eliminating E, F and H in (22) using (19)–(21), and evaluating the result we find
m σ m ρ σi ρi = α|X |m γ γi
(0 ≤ i ≤ d).
(23)
TIGHT GRAPHS AND THEIR PRIMITIVE IDEMPOTENTS
55
Setting i = 0 in (23), and recalling σ0 = 1, ρ0 = 1 and γ0 = 1, we find
−1
α = m σ m ρ m −1
γ |X | .
(24)
Eliminating α in (23) using (24), we obtain
σi ρi = γi
(0 ≤ i ≤ d),
as desired.
(ii) ⇒ (i) By (19)–(21),
E ◦ F = |X |−2 m σ m ρ
d
X
σi ρi Ai
i=0
= |X |−2 m σ m ρ
d
X
γi Ai
i=0
= α H,
where α is as in (24).
2
In the next lemma, we consider some examples.
Lemma 3.3 Let 0 denote a distance-regular graph with diameter d ≥ 3, and eigenvalues
θ0 > θ1 > · · · > θd .
(i) E 0 ◦ E i = |X |−1 E i (0 ≤ i ≤ d).
(ii) Suppose 0 is bipartite. Then
E d ◦ E i = |X |−1 E d−i (0 ≤ i ≤ d).
Proof:
(i) Follows easily from (5).
(ii) By Lemma 2.4, the cosine sequence for E d is 1, −1, 1, . . . , (−1)d . Let σ0 , σ1 , . . . , σd
denote the cosine sequence for E i . By Lemma 2.5, the cosine sequence for E d−i is
σ0 , −σ1 , σ2 , . . . , (−1)d σd . Combining the above information with Lemma 3.2, we find
E d ◦ E i is a scalar multiple of E d−i . The scalar is |X |−1 by (18), Lemma 2.5 (ii), and
the fact that m 0 = 1.
2
Theorem 3.4 Let 0 denote a distance-regular graph with diameter d ≥ 3. Let E and F
denote primitive idempotents of 0, with cosine sequences σ0 , σ1 , . . . , σd and ρ0 , ρ1 , . . . , ρd ,
respectively. The following are equivalent.
(i) E ◦ F is a scalar multiple of a primitive idempotent of 0.
(ii) There exists a real number ² such that
σi ρi − σi−1 ρi−1 = ²(σi−1 ρi − σi ρi−1 ) (1 ≤ i ≤ d).
(25)
56
PASCASIO
Proof:
(i) ⇒ (ii) Suppose E ◦ F is a scalar multiple of a primitive idempotent H of 0. Let
γ0 , γ1 , . . . , γd be the cosine sequence for H . First, assume E 6= F and set
²=
σρ − 1
.
ρ−σ
(26)
Pick an integer i (1 ≤ i ≤ d). Observe by Lemma 3.2, Corollary 2.3, Lemma 2.2 and
Eq. (26),
σi ρi − σi−1 ρi−1 = γi − γi−1
= (γ − 1)
i−1
c1 c2 · · · ci−1 X
k h γh
b1 b2 · · · bi−1 h=0
i−1
c1 c2 · · · ci−1 X
k h σh ρh
b1 b2 · · · bi−1 h=0
¶
µ
σi ρi−1 − σi−1 ρi
= (σρ − 1)
σ −ρ
= (σρ − 1)
= ²(σi−1 ρi − σi ρi−1 ),
as desired.
Next suppose E = F. Then σi = ρi for 0 ≤ i ≤ d, so in view of Lemma 3.2,
γi = σi2
(0 ≤ i ≤ d).
Observe γ0 , γ1 , . . . , γd are all non-negative, so H = E 0 by Lemma 2.6. In particular,
γi = 1 for 0 ≤ i ≤ d, so
σi2 = 1
(0 ≤ i ≤ d).
Now, (25) holds for all ² ∈ R.
(ii) ⇒ (i) Set
γi := σi ρi
(0 ≤ i ≤ d).
(27)
By Lemma 3.2, it suffices to show γ0 , γ1 , . . . , γd is a cosine sequence. By Corollary 2.3,
this will occur if we can show
(γ − 1)
i
X
h=0
k h γh =
b1 b2 · · · bi
(γi+1 − γi )
c1 c2 · · · ci
(28)
for 0 ≤ i ≤ d, where γd+1 is indeterminate. Setting i = 1 in (25), we obtain
σρ − 1 = ²(ρ − σ ).
(29)
TIGHT GRAPHS AND THEIR PRIMITIVE IDEMPOTENTS
57
By Lemma 2.2,
(σ − ρ)
i
X
k h σh ρh =
h=0
b1 b2 · · · bi
(σi+1 ρi − σi ρi+1 )
c1 c2 · · · ci
(30)
for 0 ≤ i ≤ d. We first show (28) holds at i = d. Observe that the right side of (30)
vanishes at i = d. Combining this observation with (27) and (29), we find
(γ − 1)
d
X
kh γh = (σρ − 1)
h=0
d
X
k h σh ρh
(31)
k h σh ρh
(32)
h=0
= ²(ρ − σ )
d
X
h=0
= 0,
(33)
so (28) holds at i = d.
Next, we show (28) holds for i < d. Multiplying Eq. (30) by ², and simplifying the
resulting equation using (25) and (29), we obtain
(σρ − 1)
i
X
k h σh ρh =
h=0
b1 b2 · · · bi
(σi+1 ρi+1 − σi ρi ),
c1 c2 · · · ci
(34)
which implies (28) in view of (27). We have shown (28) for 0 ≤ i ≤ d, so γ0 , γ1 , . . . , γd is
2
a cosine sequence by Corollary 2.3.
We conclude by determining which nontrivial primitive idempotents E and F satisfy (i),
(ii) of Theorem 3.4. We consider three cases.
Theorem 3.5 Let 0 denote a tight distance-regular graph with diameter d ≥ 3, and
eigenvalues θ0 > θ1 > · · · > θd . Let E and F denote nontrivial primitive idempotents of 0.
The following are equivalent.
(i) E ◦ F is a scalar multiple of a primitive idempotent of 0.
(ii) E, F are a permutation of E 1 , E d .
Suppose (i), (ii) hold, and let H denote the primitive idempotent of 0 such that E ◦ F is a
scalar multiple of H . Then the eigenvalue associated with H is θd−1 . Moreover,
kθd−1 = θ1 θd .
(35)
Proof:
(i) ⇔ (ii) Follows from Theorem 1.2 and Theorem 3.4.
If (i), (ii) hold then the eigenvalues of 0 associated with E and F are a permutation
of θ1 and θd . Let σ0 , σ1 , . . . , σd and ρ0 , ρ1 , . . . , ρd denote the cosine sequences for θ1
58
PASCASIO
and θd , respectively. By Lemma 2.6, σ0 , σ1 , . . . , σd has 1 sign change and ρ0 , ρ1 , . . . , ρd
has d sign changes. Combining this with Lemma 3.2 and Lemma 2.7, we observe that
the cosine sequence for H , namely σ0 ρ0 , σ1 ρ1 , . . . , σd ρd has d − 1 sign changes. By
Lemma 2.6, the eigenvalue associated with H is θd−1 . Finally, applying Lemma 2.1 (iii),
2
we get σ1 = θ1 k −1 , ρ1 = θd k −1 , and σ1 ρ1 = θd−1 k −1 , giving (35), as desired.
Theorem 3.6 Let 0 denote a bipartite distance-regular graph with diameter d ≥ 3, and
eigenvalues θ0 > θ1 > · · · > θd . Let E and F denote nontrivial primitive idempotents
of 0. The following are equivalent.
(i) E ◦ F is a scalar multiple of a primitive idempotent of 0.
(ii) At least one of E, F is equal to E d .
Proof:
(i) ⇒ (ii) Let σ0 , σ1 , . . . , σd and ρ0 , ρ1 , . . . , ρd denote the cosine sequences for E and F,
respectively. By Theorem 3.4, there exists a real number ² such that
σi ρi − σi−1 ρi−1 = ²(σi−1 ρi − σi ρi−1 ) (1 ≤ i ≤ d).
(36)
First assume E = F. Setting i = 1 in (36), we observe σ12 = σ02 = 1. Observe σ1 6= 1
since E is nontrivial, so σ1 = −1. Now E = E d by Lemma 2.4 (i), (ii).
Next assume E 6= F. Setting i = 1, 2 in (36), we get
σρ − 1 = ²(ρ − σ ),
(37)
σ2 ρ2 − σρ = ²(σρ2 − σ2 ρ).
(38)
Let θ and θ 0 denote the eigenvalues for E and F, respectively. Then by Lemma 2.1
(iii),
σ =
θ
,
k
ρ=
θ0
.
k
(39)
Recalling that a0 , a1 , . . . , ad are 0 since 0 is bipartite, and setting i = 1 in (6) and (3),
we get
σ2 =
θ2 − k
θ 02 − k
, ρ2 =
.
k(k − 1)
k(k − 1)
(40)
Eliminating ² in (38) using (37), and simplifying the result using (39) and (40), we
obtain
(θ 2 − k 2 )(θ 02 − k 2 ) = 0.
Observe θ 6= k and θ 0 6= k, since E and F are nontrivial, so one of θ, θ 0 is equal to −k.
Thus, one of E, F is equal to E d .
TIGHT GRAPHS AND THEIR PRIMITIVE IDEMPOTENTS
59
(ii) ⇒ (i) Follows immediately from Lemma 3.3 (ii).
2
Theorem 3.7 Let 0 denote a distance-regular graph with diameter d ≥ 3, and suppose
0 is neither tight nor bipartite. Let E and F denote nontrivial primitive idempotents of 0.
Then E ◦ F is never a scalar multiple of a primitive idempotent of 0.
Proof:
Immediate from Theorem 1.2 and Theorem 3.4.
2
Acknowledgments
The author wishes to thank Professor Paul Terwilliger for his ideas and many valuable
suggestions, and Jack Koolen for observing that H is associated to θd−1 in Theorem 3.5.
References
1. E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin-Cummings Lecture Note
Ser. 58, Benjamin-Cummings, Menlo Park, CA. 1984.
2. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer, New York, 1989.
3. C.D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993.
4. A. Juris̆ić, J. Koolen, and P. Terwilliger, “Tight distance-regular graphs,” submitted.