Partially Distance-Regular Graphs and Partially Walk-Regular Graphs.

Partially Distance-regular Graphs and

Partially Walk-regular Graphs ^∗

Tayuan Huang

Yu-pei Huang

Shu-Chung Liu

Chih-wen Weng

August 1, 2007

Abstract

We study partially distance-regular graphs and partially walk- regular graphs as generalizations of distance-regular graphs and walk- regular graphs respectively. We conclude that the partially distance- regular graphs can be viewed as some extremal graphs of partially walk-regular graphs. In the special case that the graph is assumed to be regular with four distinct eigenvalues, a well known class of walk- regular graphs, we show that there exists a rational function f in the expression of the order and the four eigenvalues of the graph such that k₂(x), the number of vertices with distance 2 to a vertex x, satisfies k₂(x) ≥ f ; furthermore we show the equality holds for each vertex x if and only if the graph is distance-regular with diameter 3.

Keywords: Partially distance-regular graphs; partially walk-regular graphs, eigenvalues.

∗Research partially supported by the NSC of Taiwan R.O.C. under projects NSC 95- 2115-M-002 and ...

†Department of Applied Mathematics, National Chiao Tung University, Taiwan R.O.C.

‡Graduate Institute of Computer Science, National Hsinchu University of Education, Taiwan R.O.C.

1 Introduction

Partially distance-regular graphs and partially walk-regular graphs (formal definition in Section 2 and Section 3) are generalizations of distance-regular graphs and walk-regular graphs respectively. They were introduced by M. A.

Fiol and E. Garriga when they studied the range of the spectrum of a graph [6].

We study these two classes of graphs and find their properties by follow- ing the classic way in the study of distance-regular graphs and walk-regular graphs respectively. We study the link between these two classes of graphs, and conclude that the partially distance-regular graphs can be viewed as some extremal graphs of partially walk-regular graphs. See Theroem 3.3 for detailed description.

We apply Theorem 3.3 to the case when the graph Γ is assumed to be regular and has exactly four distinct eigenvalues. It is well known that Γ is walk-regular. See Lemma 5.1 for the generalization of this result. We show that there exists a rational function f in the expression of the order and the four eigenvalues of Γ such that k₂(x), the number of vertices with distance 2 to a vertex x, satisfies k2(x) ≥ f. Furthermore we show the equality holds for each vertex x of Γ if and only if Γ is distance-regular with diameter 3.

See Theorem 6.1 for detail. Theorem 6.1 answers a conjecture in [4].

It is well known that the Godsil switching, a way of switching edges of a graph (described in Section 4), will destroy the distance-regularity of a graph while preserving its spectrum. We show in Corollary 4.3 that the walk- regularity of a graph is preserved by Godsil switching provided the switching exists.

2 Partially Distance-regular Graphs

Throughout the paper, let Γ = (X, R) denote a simple connected graph with diameter d, order n, and length distance function ∂. For an integer i and x ∈ X, set

Γi(x) := {y | y ∈ X, ∂(y, x) = i}.

Γ is k-regular or regular for short if |Γ1(x)| = k for all x ∈ X. The i-th distance matrix A_i is an n × n matrix with rows and columns indexed by X such that

(Ai)xy :=

½ 1, if ∂(x, y) = i;

0, else, (x, y ∈ X).

Hence A_i = 0 for i < 0 or i > d. A = A₁ is called the adjacency matrix of Γ.

Definition 2.1. We say that Γ is t-partially distance-regular whenever for each integer 0 ≤ i ≤ t, there exists a polynomial v_i(λ) ∈ IR[λ] of degree i such that A_i = v_i(A). Γ is distance-regular if Γ is d-partially distance-regular.

Hence any graph is 1-partially distance-regular from the above definition.

Definition 2.2. Fix integers 0 ≤ i, j, h ≤ d. We say P_ij^h is well-defined in Γ whenever for any two vertices x, y ∈ X with ∂(x, y) = h, the number

p^h_ij(x, y) := |Γ_i(x) ∩ Γ_j(y)|

is independent of the choice of x, y. In this case we write p^h_ij for p^h_ij(x, y), and call p^h_ij the intersection number of Γ. For convenience we will write ch, ah, bh, and k_h for the symbols p^h_{1 h−1}, p^h_{1 h}, p^h_{1 h+1} and p⁰_hh respectively.

Note that k₁(x) is the valency of x ∈ X. Observe that for h ≥ 1, a_h(x, y)+

b_h(x, y) + c_h(x, y) = k₁(x) for x, y ∈ X with ∂(x, y) = h. The following proposition is similar to a well-known property in the study of distance- regular graphs [1, Proposition 1.1].

Proposition 2.3. Fix an integer 1 ≤ t ≤ d. Then the following are equiva- lent.

(i) Γ is t-partially distance-regular.

(ii) p^k_ij are well-defined for 0 ≤ i + j, k ≤ t.

(iii) c_i, a_i−1, b_i−2 are well-defined for 1 ≤ i ≤ t.

Proof. ( (i)=⇒(ii) ) Fix two integers i, j with 0 ≤ i+j ≤ t. By the assumption (i) we know {A₀, A₁, . . . , A_t} is a basis of the vector space { f (A) | f (λ) ∈ IR[λ] has degree at most t} over IR, and A_iA_j is in the vector space. Hence

A_iA_j = Xt

k=0

c^k_ijA_k (2.1)

for some constants c^k_ij ∈ IR. For any two vertices x, y ∈ X with ∂(x, y) = k ≤ t, comparing the xy entry on both sides of (2.1), we find that p^k_ij(x, y) = c^k_ij is independent of x, y.

((ii)=⇒(iii)) For 1 ≤ i ≤ t, c_i = pⁱ_{1 i−1}, a_i−1 = pⁱ⁻¹_{1 i−1}, b_i−2 = pⁱ⁻²_{1 i−1} are well-defined, since the sum is i ≤ t in each of the subscripts of intersection numbers.

((iii)=⇒(i)) Note that

AA_i−1= b_i−2A_i−2+ a_i−1A_i−1+ c_iA_i by comparing entries on both sides, or equivalently

A_i = c⁻¹_i ((A − a_i−1I)A_i−1− b_i−2A_i−2)

for 1 ≤ i ≤ t − 1. Hence Ai = vi(A) where vi(λ) ∈ R[λ] has degree i and is defined recursively by v₀(λ) = 1, v₁(λ) = λ, and

v_i(λ) = c⁻¹_i ((λ − a_i−1)v_i−1(λ) − b_i−2v_i−2(λ)) for 2 ≤ i ≤ t.

It is not clear at this moment that k_t= p⁰_tt is well-defined from Proposi- tion 2.3(ii). The following lemma explains this.

Lemma 2.4. Suppose Γ is t-partially distance-regular, where t ≥ 2. Then bt−1 and ki are well-defined for 0 ≤ i ≤ t.

Proof. We apply Proposition 2.3. Note b₀ is well-defined since t ≥ 2. Since bt−1 = b0 − at−1 − ct−1, we find bt−1 is well-defined. As in [2, Chapter 5], we have k_i = b₀b₁· · · b_i−1/(c₁c₂· · · c_i), and hence k_i is well-defined for 0 ≤ i ≤ t.

3 Partially Walk-regular Graphs

We now give the definition of the second class of graphs in the title of the paper.

Definition 3.1. We say that Γ is t-partially walk-regular whenever for each integer 1 ≤ i ≤ t, (Aⁱ)xx is a constant depending on i, but not on x ∈ X. Γ is walk-regular if Γ is t-partially walk-regular for any integer t.

Hence in a t-partially walk-regular graph, the number of closed walks of length i from a vertex x to itself is a constant, depending on i ≤ t not on x ∈ X. In particular, a 2-partially walk-regular graph is regular with valency (A²)xx for any x ∈ X.

Lemma 3.2. Suppose Γ is t-partially distance-regular, where t ≥ 2. Then Γ is 2t-partially walk-regular.

Proof. Fix a positive integer u ≤ t. Suppose

A^u−1= Xu−1

i=0

t_iA_i, A^u = Xu

i=0

s_iA_i

for some t_i, s_i ∈ IR. Note that k_i is well-defined for 0 ≤ i ≤ t by Lemma 2.4.

Then

(A^2u−1)_xx = X

y∈X

(A^u−1)_xy(A^u)_yx

= X

y∈X

( Xu−1

i=0

t_i(A_i)_xy)(

Xu i=0

s_i(A_i)_xy)

= Xu−1

i=0

k_it_is_i,

and

(A^2u)_xx = X

y∈X

( Xu

i=0

s_i(A_i)_xy)²

= Xu

i=0

kis²_i

are independent of the choice of x ∈ X.

The converse of the above lemma is false. C6, the complement of a cycle of length 6, is a graph of diameter 2, which is walk-regular, but not distance- regular.

Theorem 3.3. Suppose Γ is regular and t-partially distance-regular. Then for x ∈ X, we have |Γ_t+1(x)| ≥ f, where f is a function of intersection

numbers and (A^2t+1)xx, (A^2t+2)xx. Furthermore suppose Γ is (2t + 2)-partially walk-regular. Then the above equality holds for each x ∈ X if and only if Γ is (t + 1)-partially distance-regular.

Proof. If Γt+1(x) = ∅, we choose f = 0 and the first part of the theorem holds clearly. We assume Γ_t+1(x) 6= ∅. By the assumption we can write A^t = P^t

i=0

s_iA_i for some s_i ∈ IR with s_t 6= 0. Also c_i, b_i−1, a_i−1 and k_i are well-defined in Γ for 1 ≤ i ≤ t by Proposition 2.3 and Lemma 2.4. Then

(A^2t+1)_xx = X

y∈X

(X

z∈X

Xt i=0

s_i(A_i)_xzA_zy)(

Xt i=0

s_i(A_i)_yx)

= Xt−1

i=0

k_i(c_is_i−1+ a_is_i+ b_is_i+1)s_i + (k_tc_ts_t−1+ X

y∈Γt(x)

a_t(y, x)s_t)s_t (3.1)

by summing y according to its distance i to x. From (3.1) we find X

y∈Γt(x)

at(y, x)

can be determined from the well-defined intersection numbers of Γ and an additional constant (A^2t+1)_xx. Similarly,

(A^2t+2)xx = X

y∈X

((A^t+1)xy)²

= X

y∈X

(X

z∈X

Xt i=0

s_i(A_i)_xz(A)_zy)²

= Xt−1

i=0

k_i(c_is_i−1+ a_is_i+ b_is_i+1)² (3.2)

+ X

y∈Γt(x)

(ctst−1+ at(y, x)st)² (3.3)

+ X

y∈Γt+1(x)

(c_t+1(y, x)s_t)². (3.4)

By applying Cauchy’s inequality in (3.3), X

y∈Γt(x)

(c_ts_t−1+ a_t(y, x)s_t)²

≥ 1

k_t( X

y∈Γt(x)

(c_ts_t−1+ a_t(y, x)s_t))². (3.5)

and equality holds in (3.5) iff a_t(y, x) is independent of the choice of y ∈ Γ_t(x).

Similarly for (3.4) we have X

y∈Γt+1(x)

(c_t+1(y, x)s_t)²

≥ 1

k_t+1(x)( X

y∈Γt+1(x)

c_t+1(y, x)s_t)²

= 1

kt+1(x)( X

y∈Γt(x)

(b₀− c_t− a_t(y, x))s_t)², (3.6)

and equality holds iff c_t+1(y, x) is independent of the choice y ∈ Γ_t+1(x).

Set T1, T2, (kt+1(x))⁻¹T3 to be the expressions in (3.2), (3.5) and (3.6) re- spectively, and note that T₁, T₂, T₃ can be computed from the intersection numbers of Γ and the additional constant (A^2t+1)_xx. Now we have

(A^2t+2)xx ≥ T1+ T2+ (kt+1(x))⁻¹T3.

Note that T₃ > 0. Then (A^2t+2)_xx− T₁− T₂ > 0. So we can rewrite the above inequality as

k_t+1(x) ≥ T₃

(A^2t+2)xx− T1− T2

. (3.7)

The first part of the theorem is obtained by setting f to be the right hand side of (3.7).

Suppose Γ is (2t+2)-partially walk-regular. Then the f is a constant, not depending on x ∈ X. We consider two cases according to f = 0 or not. Note that f = |Γt+1(x)| = 0 for all x ∈ X iff d ≤ t, and this is equivalent to that Γ is distance-regular. Suppose |Γ_t+1(x)| 6= 0 for some x ∈ X. Then the equality hold in (3.7) for each x ∈ X iff c_t+1, a_t, b_t = b₀− c_t− a_t are well-defined, and this is equivalent to that Γ is (t + 1)-partially distance-regular.

Remark 3.4. The inequality in Theorem 3.3 essentially comes from Cauchy’s inequality. A similar argument also appears in [5].

4 Godsil Switching

We shall prove the walk-regularity are preserved by two operations on graphs in this section.

The complement Γ of Γ = (X, R) is a graph with vertex set X and adjacency matrix A = J − I − A, where A is the adjacency matrix of Γ, I is the identity matrix and J is the all 1’s matrix.

Lemma 4.1. Suppose Γ = (X, R) is t-partially walk-regular. Then the com- plement Γ of Γ is t-partially walk regular.

Proof. This is clear if t ≤ 1 since every graph is 1-partially walk-regular.

Assume t ≥ 2. In particular Γ is k-regular for some nonegative integer k.

Since JA = AJ = kJ and JJ = nJ, we find A ⁱ = (J − I − A)ⁱ is a linear combination of J, I, A; in particular A ⁱ has identical diagonal entries for 0 ≤ i ≤ t.

Suppose π = (C₁, C₂, . . . , C_k, C_k+1) is a partition of the vertex set X such that the following (i)-(ii) hold.

(i) For 1 ≤ i, j ≤ k, there exists a constant t_ij such that

|Γ₁(x) ∩ C_j| = t_ij (4.1) for all x ∈ C_i.

(ii) For x ∈ C_k+1 and 1 ≤ i ≤ k,

|Γ₁(x) ∩ C_i| = 0, |C_i|/2, or |C_i|.

Suppose the above partition π exists in Γ = (X, R). Let Γ^(π) denote the graph with same vertex set X and the same edges of Γ except that for each x ∈ Ck+1 and each 1 ≤ i ≤ k with |Γ1(x) ∩ Ci| = |Ci|/2, the edges between x and C_i are deleted and all the edges from x to vertices in C_i− Γ₁(x) are added. We say the graph Γ^(π) is obtained from Γ by the Godsil switching with respect to π. To describe the adjacency matrix A^(π) of Γ^(π) as shown in [7], we need the following setting. For positive integers m, t, let J_m (resp.

j_m) denote the m × m (resp. m × 1) all 1’s matrix, I_m denote the m × m identity matrix and Qm = (2/m)Jm− Im. The the following (a)-(d) are easily verified.

(a) Q²_m = Im,

(b) If X is an m×t matrix with a constant row sum and a constant column sum, then Q_mXQ_t= X,

(c) If X is an m × 1(1 × m) matrix with column sum (row sum) m/2 , then Q_mX = j_m− X ((j_m)^T − X = XQ_m),

(d) Q_mj_m = j_m.

We may assume that the vertices of Γ are ordered so that A can be written

as 





B11 B12 · · · B1k+1

B₂₁ B₂₂ · · · B_2k+1 ... ... . .. ...

B_{k+1 1} B_{k+1 2} · · · B_k+1k+1





,

where B_ii is the adjacency matrix of the graph induced by C_i. Note that B_ij has a constant row sum and a constant column sum for each pair 1 ≤ i, j ≤ k.

The partition π = (C₁, C₂, . . . , C_k, C_k+1) of X is equitable if (4.1) holds for 1 ≤ i, j ≤ k + 1; in this case B_ij has a constant row sum and a constant column sum for each pair 1 ≤ i, j ≤ k + 1. Let Q be the block diagonal matrix with k + 1 blocks, where the i-th diagonal block is Q_mi if i ≤ k and the (k +1)-th block is the identity matrix I_mk+1, with m_i = |C_i|. From (a)-(d) above and the constriction, we have Q² = I and

A^(π)= QAQ =









B₁₁ B₁₂ · · · B_1k Q_m1B_1k+1 B21 B22 · · · B2k Qm2B2k+1

... ... . .. ... ...

B_k1 B_k2 B_kk Q_mkB_kk+1

B_{k+1 1}Q_m1 B_{k+1 2}Q_m2 · · · B_k+1kQ_mk B_k+1k+1







 .

Suppose A^s is written in the block matrix form as

A^s =







B₁₁^(s) B₁₂^(s) · · · B_1k+1^(s) B₂₁^(s) B₂₂^(s) · · · B_2k+1^(s)

... ... . .. ...

B_{k+1 1}^(s) B_{k+1 2}^(s) · · · B_k+1k+1^(s)





 (4.2)

for any nonnegative integer s, where B_ij⁽¹⁾ = B_ij for 1 ≤ i, j ≤ k + 1.

Proposition 4.2. Let Γ = (X, R) be a regular graph, and let π be an equitable partition of X satisfying (ii) above. Fix a nonnegative integer s and suppose A^s is as in (4.2). Then

(A^(π))^s =











B₁₁^(s) B^(s)₁₂ · · · B_1k^(s) Q_m1B_1k+1^(s) B₂₁^(s) B^(s)₂₂ · · · B_2k^(s) Q_m2B_2k+1^(s)

... ... . .. ... ...

B_k1^(s) B^(s)_k2 B_kk^(s) Q_mkB_kk+1^(s) B_{k+1 1}^(s) Q_m1 B_{k+1 2}^(s) Q_m2 · · · B_k+1k^(s) Q_mk B_k+1k+1^(s)









 .

Proof. The B_ij described above has a constant row sum and a constant col- umn sum for each pair 1 ≤ i, j ≤ k + 1, since π is equitable. This implies that

B_ij^(s)= X

1≤p1,p2,...,ps−1≤k+1

B_ip1B_p1p2· · · B_ps−2ps−1B_ps−1j

has a constant row sum and a constant column sum. Applying the above (b) to (A^(π))^s = QA^sQ and simplifying, we have the proposition.

Corollary 4.3. Let Γ = (X, R) denote a t-partially walk-regular graph, and let π be an equitable partition of X satisfying (ii) above. Then Γ^(π) is t- partially walk-regular.

Proof. The corollary follows immediately from Proposition 4.2 since A^s and (A^(π))^s have the same diagonal blocks for any nonnegative integer s.

Remark 4.4. ([8]) The Gosset graph Γ = (X, R) is the unique distance- regular graph on 56 vertices of diameter 3 with b0 = 27, b1 = 10, b2 = 1, c₂ = 10, and c₃ = 27. There exists an equitable partition π of X that satisfies (ii) above. The graph Γ^(π) obtained from Γ by Godsil switching with respect to π is not distance-regular.

5 Graphs with s Distinct Eigenvalues

In this section we assume Γ = (X, R) is a simple connected k-regular graph with diameter d, s distinct eigenvalues

k > λ₁ > λ₂ > . . . > λ_s−1,

and order n. It is well-known that s ≥ d + 1 [7, Lenna 5.2], and J = n

q(k)q(A), (5.1)

where q(λ) :=^s−1Q

i=1

(λ − λ_i) ∈ IR[λ] [3, Corollary 3.3].

Lemma 5.1. Suppose that Γ is (s − 2)-partially walk-regular, where s ≥ 4.

Then Γ is walk-regular. In particular, for any nonnegative integers t, (A^t)xx

is determined by n and the eigenvalues of Γ for all x ∈ X.

Proof. For s ≥ 4, Γ is regular. Note that the q(λ) of Γ has degree s − 1 and k^tJ = n

q(k)q(A)A^t

for all nonnegative integers t, where k is the valency of Γ. Hence (A^s−1+t)xx

is a function of k = (A²)_xx, (A³)_xx, . . . , (A^s−2)_xx for all nonnegative integers t.

Lemma 5.2. Suppose that Γ is (d − 1)-partially distance-regular and has d + 1 distinct eigenvalues, where d ≥ 3. Then Γ is distance-regular.

Proof. Γ is regular since d ≥ 3. The q(λ) of Γ has degree d since Γ has d + 1 eigenvalues. Hence by referring to (5.1), for any two vertices x, y ∈ X with

∂(x, y) = d,

(A^d)xy = q(k) n is independent of the choice of x, y, and note that

(A^d)xy = cd(y, x)cd−1cd−2· · · c1.

Hence cd is well-defined. Similarly, for any two vertices x, y ∈ X with

∂(x, y) = d − 1, (A^d)xy = q(k)

n − cd−1cd−2· · · c1× the coefficient of λ^d−1 in q(λ) is independent of the choice of x, y, and note that

(A^d)xy = (ad−1(y, x) + ad−2+ . . . + a1)(cd−1cd−2· · · c1).

Hence a_d−1 is well-defined. Since Γ is regular with diameter d, we find c_d, a_d−1, b_d−2 are well-defined.

6 Regular Graphs with Four Eigenvalues

We apply the previous results to the connected regular graphs with four distinct eigenvalues in this section.

Theorem 6.1. Let Γ = (X, R) denote a connected k-regular graph with n vertices and four distinct eigenvalues k > λ₁ > λ₂ > λ₃. Then Γ is walk-regular with diameter 2 or 3. Moreover there exists a rational function f (n, k, λ1, λ2, λ3) in the variables n, k, λ1, λ2, λ3 such that for any x ∈ X

k2(x) ≥ f (n, k, λ1, λ2, λ3). (6.1) Furthermore the equality holds for each x ∈ X if and only if Γ is distance- regular with diameter 3.

Proof. Γ is walk-regular by Lemma 5.1 and clear to have diameter 2 or 3.

It is well known that if Γ has diameter 2 then it is not distance-regular[7, Lemma 4.1]. Now the theorem follows from Theorem 3.3 with the case t = 1.

Remark 6.2. The second part of Theorem 6.1 is essentially a result of E.

R. van Dam and W. H. Haemers [4] with slightly different variables in the expression of f [4]. The inequality (6.1) is also obtained there with other additional assumptions. They conjectured these additional assumptions can be removed. Theorem 6.1 fulfills their conjecture.

References

[1] E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park, 1984.

[2] A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989.

[3] N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cam- bridge, 1993.

[4] E. R. van Dam and W. H. Haemers, A Characterization of Distance- Regular Graphs with Diameter Three, Journal of Algebraic Combina- torics, 6(1997), 299-303.

[5] E. R. van Dam and W. H. Haemers, Spectral Characterizations of Some Distance-Regular Graphs, Journal of Algebraic Combinatorics, 15(2002), 189-202.

[6] M. A. Fiol and E. Garriga, The alternating and adjacency polynomials, and their relation with the spectra and diameters of graphs, Discrete Appl. Math., 87 (1998), no. 1-3, 77–97.

[7] C. D. Godsil, Algebraic Combinatorics Chapman and Hall Mathematics, New York, 1993.

[8] W. H. Haemers, Distance-regularity and the spectrum of graphs, Linear Alg. Appl., 236(1996), 265–278.

Chih-wen Weng

Department of Applied Mathematics National Chiao Tung University 1001 Ta Hsueh Road

Hsinchu, Taiwan 300, R.O.C.

Email: [email protected] Fax: +886-3-5724679