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More circulant graphs exhibiting pretty good state transfer
Discrete Mathematics 341 (2018) 889–895
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More circulant graphs exhibiting pretty good state transfer Hiranmoy Pal Department of Mathematics, Indian Institute of Information Technology Bhagalpur, Bihar, 813210, India
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Article history: Received 27 May 2017 Received in revised form 2 November 2017 Accepted 18 December 2017 Available online 17 January 2018 Keywords: Graph Circulant graph Pretty good state transfer
a b s t r a c t The transition matrix of a graph G corresponding to the adjacency matrix A is defined by H(t) := exp (−itA) , where t ∈ R. The graph is said to exhibit pretty good state transfer between a pair of vertices u and v if there exists a sequence {tk } of real numbers such that limk→∞ H(tk )eu = γ ev , where γ is a complex number of unit modulus. We present a class of circulant graphs admitting pretty good state transfer. Also we find some circulant graphs not exhibiting pretty good state transfer. This generalizes several pre-existing results on circulant graphs admitting pretty good state transfer. © 2017 Elsevier B.V. All rights reserved.
1. Introduction In [9], Godsil first introduced the notion of pretty good state transfer (PGST) in the context of continuous-time quantum walks. State transfer has significant applications in quantum information processing and cryptography (see [4,6]). We consider state transfer with respect to the adjacency matrix of a graph. It can be considered with respect the Laplacian matrix as well. However, for a regular graph G, one can observe that G admits PGST with respect to the adjacency matrix if and only if G admits PGST with respect to the Laplacian matrix. All graphs we consider are simple, undirected and finite. Now we define PGST on a graph G with adjacency matrix A. The transition matrix of G is defined by H(t) := exp (−itA) , where t ∈ R. Let eu denote the characteristic vector corresponding to the vertex u of G. The graph G is said to exhibit PGST between a pair of vertices u and v if there is a sequence {tk } of real numbers such that lim H(tk )eu = γ ev , where γ ∈ C and |γ | = 1.
k→∞
In the past decade the study of state transfer has received considerable attention. Now we know a few graphs exhibiting PGST. The main goal here is to find new graphs that admits PGST. However it is more desirable to find PGST in graphs with large diameters. In [10], Godsil et al. established that the path on n vertices, denoted by Pn , exhibits PGST between the end vertices if and only if n + 1 equals to either 2m or p or 2p, where p is an odd prime. Further investigations have been done in [5] on the existence of PGST in internal vertices of some paths Pn . In [20], van Bommel established a complete characterization of PGST in Pn between any pair of vertices. A complete characterization of cycles exhibiting PGST appears in [15]. There we see that a cycle on n vertices exhibits PGST if and only if n is a power of 2. In [7], Fan and Godsil showed that a double star Sk,k admits PGST if and only if 4k + 1 is not a perfect square. A NEPS is a graph product which generalizes several well known graph products, namely, Cartesian product, Kronecker product etc. In [16], we see some NEPS of the path on three vertices having PGST. Also, in [1,2], PGST has been studied on corona products. Some other relevant results regarding state transfer can be found in [9,13,17,14,18]. In the present article, we find a class of circulant graphs admitting PGST. Also we find some E-mail address: [email protected]. https://doi.org/10.1016/j.disc.2017.12.012 0012-365X/© 2017 Elsevier B.V. All rights reserved.
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circulant graphs not exhibiting pretty good state transfer. The results established in this article generalize several results appearing in [15] on circulant graphs admitting PGST. Let (Γ , +) be a finite abelian group and consider S \ {0} ⊆ Γ with {−s : s ∈ S } = S. A Cayley graph over Γ with the symmetric set S is denoted by Cay (Γ , S ). The graph has the vertex set Γ where two vertices a, b ∈ Γ are adjacent if and only if a − b ∈ S. The set S is called the connection set of Cay (Γ , S ). Let Zn be the cyclic group of order n. A circulant graph is a Cayley graph over Zn . The cycle Cn , in particular, is the circulant graph over Zn with the connection set {1, n − 1}. The eigenvalues and its corresponding eigenvectors of a cycle are very well known. The eigenvalues of Cn are
λl = 2 cos
(
2lπ
)
n
, 0 ≤ l ≤ n − 1,
(1)
[
l(n−1)
and the corresponding eigenvectors are vl = 1, ωnl , . . . , ωn
]T
, where ωn = exp
( 2π i ) n
is the primitive nth root of unity.
In general, the eigenvalues and eigenvectors of a Cayley graph over an abelian group are also known in terms of characters of the abelian group. In [11], it appears that the eigenvectors of a Cayley graph over an abelian group are independent of the connection set. Therefore the sets of eigenvectors of two Cayley graphs defined over an abelian group can be chosen to be equal. Hence we have the following result. Proposition 1.1. If S1 and S2 are symmetric subsets of an abelian group Γ then adjacency matrices of the Cayley graphs Cay(Γ , S1 ) and Cay(Γ , S2 ) commute. Let G and H be two simple graphs on the same vertex set V , and the respective edge sets E(G) and E(H). The edge union of G and H, denoted G ∪ H, is defined on the vertex set V whose edge set is E(G) ∪ E(H). Using Proposition 1.1 and the exponential nature of the transition matrix, we obtain the following result which allows us to find the transition matrix of an edge union of two edge disjoint Cayley graphs. Proposition 1.2 ([17]). Let Γ be a finite abelian group and consider two disjoint and symmetric subsets S , T ⊂ Γ . Suppose the transition matrices of Cay(Γ , S) and Cay(Γ , T ) are HS (t) and HT (t), respectively. Then Cay(Γ , S ∪ T ) has the transition matrix HS (t)HT (t). Let n ∈ N and d be a proper divisor of n. We denote Sn (d) = {x ∈ Zn : gcd(x, n) = d} , and for any set D of proper divisors of n, we define Sn (D) =
⋃
Sn (d).
d∈D
The set Sn (D) is called a gcd-set of Zn . A graph is called integral if all its eigenvalues are integers. The following theorem determines which circulant graphs are integral. Theorem 1.3 ([19]). A circulant graph Cay (Zn , S ) is integral if and only if S is a gcd-set. It is therefore clear that if a circulant graph Cay (Zn , S ) is non-integral then there exists a proper divisor d of n such that S ∩ Sn (d) is a non-empty proper subset of Sn (d). A graph G is said to exhibit perfect state transfer (PST) between a pair of vertices u and v if there exists t ∈ R such that H(t)eu = γ ev , where γ ∈ C and |γ | = 1. The graph G is said to be periodic at a vertex u if there is t(̸ = 0) ∈ R such that H(t)eu = γ eu , where γ ∈ C and |γ | = 1. Moreover, the graph G is said to be periodic if it is periodic at all vertices at the same time. More information regarding periodicity and PST can be found in [8]. It can be observed that PGST is a generalization to PST. However, in [12], we observe in Proposition 1.3.1 that if a graph is periodic then it admits PST if and only if it admits PGST. It is worth mentioning the fact presented in [18] that a circulant graph is periodic if and only if the graph is integral. In [3], Bašić considered integral circulant graphs and presented a complete characterization of integral circulant networks exhibiting PST. This motivates us to study non-integral circulant graphs for PGST. 2. Pretty good state transfer on circulant graphs Let us recall some important observations on circulant graphs exhibiting PGST (see [15]). We include the following discussion for convenience. Suppose G is a graph with adjacency matrix A. It is well known that the matrix P of an automorphism of G commutes with A. By the spectral decomposition of the transition matrix H(t) of G, we conclude that H(t) is a polynomial in A. Therefore P commutes with H(t) as well. Suppose G admits PGST between two vertices u and v . Then there exists a sequence {tk } of real numbers and γ ∈ C with |γ | = 1 such that lim H(tk )eu = γ ev .
k→∞
H. Pal / Discrete Mathematics 341 (2018) 889–895
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Further this implies that lim H(tk ) (Peu ) = γ (Pev ) .
k→∞
The sequence {H(tk )eu } cannot have two different limits, and therefore if P fixes eu then P must also fix ev . For the circulant graph Cay (Zn , S ), we know that the map sending j to n − j is an automorphism. As a consequence, we have the following result. Lemma 2.1 ([15]). If pretty good state transfer occurs in a circulant graph Cay (Zn , S ) then n is even and it occurs only between the pair of vertices u and u + 2n , where u, u + 2n ∈ Zn . A graph G is called vertex transitive if for any pair of vertices u, v in G there exists an automorphism sending u to v . All circulant graphs are well known to be vertex transitive. Thus, by Lemma 2.1, it is enough to find PGST in a circulant graph between the pair of vertices 0 and 2n . First we calculate the (0, 2n )th entry of the transition matrix of the cycle Cn . ∑n−1 1 ∗ Suppose A = l=0 λl El is the spectral decomposition of the adjacency matrix of Cn . Here notice that El = n vl vl , where
[
l(n−1)
vl = 1, ωnl , . . . , ωn
]T
as given in (1). Now we obtain the transition matrix of Cn as
H(t) = exp (−itA) =
n−1 ∑
exp (−iλl t ) El .
l=0
− nl 2
Since (0, 2n )th entry of El is 1n ωn H(t)0, n = 2
n−1 1∑
n
, we find the (0, 2n )th entry of H(t) as − nl 2
exp (−iλl t ) · ωn
=
l=0
n−1 1∑
n
exp [−i (λl t + lπ)] .
(2)
l=0
The next result finds some cycles admitting PGST with respect to a sequence in 2π Z. Later we extend this result to more general circulant graphs. Lemma 2.2 ([15]). A cycle Cn exhibits pretty good state transfer if n = 2k , k ≥ 2, with respect to a sequence in 2π Z. A graph G is said to be almost periodic if there is a sequence {tk } of real numbers and a complex number γ of unit modulus such that limk→∞ H(tk ) = γ I , where I is the identity matrix of appropriate order. Since the circulant graphs are vertex transitive, we observe that a circulant graph is almost periodic if and only if limk→∞ H(tk )e0 = γ e0 . This implies that if a cycle admits PGST then it is necessarily almost periodic. Hence, by Lemma 2.2, all cycles of size n = 2k , k ≥ 2, are almost periodic. However, in the next result, we find a connection between the time sequences of two cycles one of which exhibits PGST and the other one is almost periodic. Lemma 2.3. Let k, k′ ∈ N and 2 ≤ k′ < k. If the cycle C2k admits pretty good state transfer with respect to a sequence {tm } then C2k′ is almost periodic with respect to a sub-sequence of {tm }. Proof. We begin with the observation that if k′ < k then
( cos
2lπ
⎛ ( = cos ⎝
′
2k
′
2 l · 2k−k
)
2k
) ⎞ π ⎠.
Consequently, if θl ’s and λl ’s are the eigenvalues of C2k′ and C2k , respectively, then we necessarily have θl = λl·2k−k′ , where ′ 0 ≤ l ≤ 2k − 1. Suppose C2k admits PGST with respect to a time sequence {tm }. Therefore, by Eq. (2), there exists a complex number γ with |γ | = 1 such that 2k −1
lim
∑
m→∞
exp [−i (λl tm + lπ )] = 2k γ .
l=0
{ }
′ Since the unit circle is compact, we have a subsequence tm of {tm } such that ′ lim exp[−i λl tm + lπ ] = γ , for 0 ≤ l ≤ 2k − 1.
(
)
m→∞
This further gives ′
′ lim exp[−i θl tm ] = γ , for 0 ≤ l ≤ 2k − 1.
(
m→∞
)
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Hence we have ′ 2k −1
lim
∑
m→∞
′ exp −i θl tm
[
(
)]
′
= 2k γ .
(3)
l=0
Notice that the (0, 0)th entry of the transition matrix of C2k′ is
{′}
1 ′ 2k
∑2k′ −1 l=0
exp [−i (θl t )] , where t ∈ R. Therefore, by Eq. (3),
we conclude that C2k′ is almost periodic with respect to tm . Hence we have the desired result. □ Now we extend Lemma 2.2 to obtain more circulant graphs (apart from the cycles) on 2k vertices exhibiting PGST. Also we find some circulant graphs which are almost periodic. It can well be observed that Theorem 7 appearing in [15] becomes a special case to the following theorem. Theorem 2.4. Let k ∈ N and n = 2k . Also let Cay (Zn , S ) be a non-integral circulant graph. Let d be the least among all the divisors of n so that S ∩ Sn (d) is a non-empty proper subset of Sn (d). If |S ∩ Sn (d)| ≡ 2 (mod 4) then Cay (Zn , S ) admits pretty good state transfer with respect to a sequence in 2π Z. Moreover, if |S ∩ Sn (d)| ≡ 0 (mod 4) then Cay (Zn , S ) is almost periodic with respect to a sequence in 2π Z. Proof. ( For each x ∈)S ∩ Sn (d), observe that Cay (Zn , {x, n − x}) is isomorphic to the disjoint( union of d copies ) of the cycle Cay Z n , 1,
} { } − 1 . By Lemma 2.2, there exists a sequence {tm } in 2π Z such that Cay Z nd , 1, nd − 1 admits PGST ( ( ) ) nx n with respect to {tm }. In particular, the cycle 0, x, . . . , 2d , . . . , nd − 1 x admits PGST between the pair of vertices 0 and 2d nx nx n between the pair of vertices 0 and 2d . Note that 2d ≡ 2 (mod n). This in turn implies that for all x ∈ S ∩ Sn (d), the graph Cay (Zn , {x, n − x}) exhibits PGST between the pair of vertices 0 and 2n with respect to {tm }. If HS ∩Sn (d) (t) is the transition matrix of Cay (Zn , S ∩ Sn (d)) then by Proposition 1.2, we can express HS ∩Sn (d) (t) as ∏ Hx (t), (4) HS ∩Sn (d) (t) = {
d
n d
x∈S ∩Sn (d), x< 2n
where Hx (t) denotes the transition matrix of Cay (Zn , {x, n − x}) for all x ∈ S ∩ Sn (d). In Eq. (4), notice that if |S ∩ Sn (d)| ≡ 2 (mod 4) then there are odd number of matrices in the product. Also we have just concluded that for all x ∈ S ∩ Sn (d) lim Hx (tm )e0 = γ e n , |γ | = 1.
(5)
2
m→∞
Since transition matrices are symmetric, we conclude that lim Hx (tm )e n = γ e0 , ∀x ∈ S ∩ Sn (d).
(6)
2
m→∞
After successive application of (5) and (6), we obtain the following from Eq. (4).
⎡ lim HS ∩Sn (d) (tm )e0 = lim ⎣
m→∞
m→∞
⎤ ∏
Hx (tm )⎦ e0 = γ
|S ∩Sn (d)| 2
en . 2
(7)
x∈S ∩Sn (d), x< 2n
′ Let S ′ = x ∈ S | gcd(x, n) = d′ and ( Sd′)⊂ S . Notice that if S is non-empty then it is in fact a gcd-set. Therefore, ( Theorem ) 1.3 implies that the sub-graph Cay Zn , S ′ of Cay (Zn , S ) is integral. Suppose HS ′ (t) is the transition matrix of Cay Zn , S ′ . Using spectral decomposition of HS ′ (t), we deduce that
{
}
HS ′ (t) = I , t ∈ 2π Z.
(8)
) Now consider S = S \ S ∪ (S ∩ Sn (d)) . By the assumption on the divisor d of n and the choice of S ′′ , we must have ′′ gcd(s, n) > d for all s ∈ S . This in turn implies that the )size of each disjoint appearing in Cay (Zn , {s, n − s}) is ( ( cycles { } { }) strictly less than the size of the cycle Cay Z n , 1, nd − 1 . Recall that Cay Z n , 1, nd − 1 admits PGST with respect d d to the sequence {tm }. Therefore, by Lemma 2.3, we have a subsequence of {tm } with respect to which each component ′′ of Cay (Zn , {s, n − s}) is almost periodic. { } Hence Cay (Zn , {s, n − s}) is ′′also almost periodic. Since S is finite, we can { } { }) appropriately choose a subsequence t of t such that for all s ∈ S , the graph Cay Z , s , n − s is{ almost ( mr m n { } ( ) } periodic ′′ with respect to tmr . Hence, by Proposition 1.2, the graph Cay Z , S is almost periodic with respect to t n mr . Therefore, ( ) if HS ′′ (t) is the transition matrix of Cay Zn , S ′′ then we have ( ) lim HS ′′ tmr e0 = ζ e0 , |ζ | = 1. (9) r →∞ { } { } Since tmr is a subsequence of {tm }, notice that Cay (Zn , S ∩ Sn (d)) also admits PGST with respect to tmr . Therefore, from (
′′
′
(7), we get lim HS ∩Sn (d) tmr e0 = γ
(
r →∞
)
|S ∩Sn (d)| 2
en . 2
(10)
H. Pal / Discrete Mathematics 341 (2018) 889–895
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Note that S ∩ Sn (d), S ′ and S ′′ are disjoint subsets of S, and S = (S ∩ Sn (d)) ∪ S ′ ∪ S ′′ . Therefore, if H(t) is the transition matrix of Cay (Zn , S ) then by Proposition 1.2, using (8), (9) and (10), we obtain
(
)
[
(
)
(
)
(
lim H tmr e0 = lim HS ∩Sn (d) tmr · HS ′′ tmr · HS ′ tmr
r →∞
)]
r →∞
e0
= lim HS ∩Sn (d) tmr · lim HS ′′ tmr e0 , as tmr ∈ 2π Z r →∞ ( ) r →∞ = lim HS ∩Sn (d) tmr · ζ e0
(
r →∞ |S ∩Sn (d)|
= ζγ
2
)
)
(
en . 2
This proves our claim that Cay (Zn , S ) admits PGST with respect to a sequence in 2π Z. For the second part of the proof, observe that, if we consider |S ∩ Sn (d)| ≡ 0 (mod 4) then we get the following equation instead of (7).
⎡
⎤ ∏
lim HS ∩Sn (d) (tm )e0 = lim ⎣
m→∞
m→∞
Hx (tm )⎦ e0 = γ
|S ∩Sn (d)| 2
e0 .
x∈S ∩Sn (d), x< 2n
Then imitating the rest of the proof of the first part, we obtain the desired result. □ We illustrate Theorem 2.4 by the following example. Example 2.1. Let n = 16 and consider Cay (Z16 , S ) where S = {1, 2, 3, 5, 11, 13, 14, 15}. Here S ∩ S16 (1) = {1, 3, 5, 11, 13, 15} ⊊ S16 (1) and S ∩ S16 (2) = {2, 14} ⊊ S16 (2). In this case, we have d = 1. Notice that |S ∩ S16 (1)| ≡ 2 (mod 4). Hence, by Theorem 2.4, that Cay (Z16 , S ) admits PGST with respect to a sequence in 2⏐π Z. Observe ⏐ that if we consider the ( we conclude ) graph Cay Z16 , S ′ with S ′ = {(1, 2, 3, )13, 14, 15}, then proceeding as above, we obtain ⏐S ′ ∩ S16 (1)⏐ ≡ 0 (mod 4). Hence, by Theorem 2.4, we find that Cay Z16 , S ′ is almost periodic with respect to a sequence in 2π Z. Now we find even more circulant graphs that is either almost periodic or admits PGST. Corollary 2.5. Suppose all the conditions of Theorem 2.4 are satisfied. Let q ∈ 2N + 1, N = nq and D be any set of proper divisors of N such that qS ∩ SN (D) = ∅. If |S ∩ Sn (d)| ≡ 2 (mod 4) then Cay (ZN , qS ∪ SN (D)) admits PGST with respect to a sequence in 2π Z. Moreover, if |S ∩ Sn (d)| ≡ 0 (mod 4) then Cay (ZN , qS ∪ SN (D)) is almost periodic with respect to a sequence in 2π Z. Proof. Note that SN (D) is a gcd-set of ZN . Therefore, by Theorem 1.3, the graph Cay (ZN , SN (D)) is integral. If HSN (D) (t) be the transition matrix of Cay (ZN , SN (D)) then using spectral decomposition of HSN (D) (t), we deduce that HSN (D) (t) = I , t ∈ 2π Z.
(11)
Let H matrix of Cay (ZN , qS ). Since qS ∩ SN (D) = ∅, by Proposition 1.2, we find the transition matrix of ( qS (t) be the transition ) Cay ZN , qS ∪ Sq (D) as H(t) = HqS (t) · HSN (D) (t), t ∈ R. If t ∈ 2π Z then using (11), we get H(t) = HqS (t).
(12)
It can be observed that Cay (ZN , qS ) is isomorphic to q disjoint copies of Cay (Zn , S ). By Theorem 2.4, if |S ∩ Sn (d)| ≡ 2 (mod 4) then Cay (Zn , S ) admits PGST with respect to a sequence in 2π Z. Hence Cay (ZN , qS ) admits PGST with respect to a sequence in 2π Z as well. Similarly, if |S ∩ Sn (d)| ≡ 0 (mod 4) then Cay (ZN , qS ) is almost periodic with respect to a sequence in 2π Z. Finally, by (12), we have the desired result. □ We present the following example supporting Corollary 2.5. Example 2.2. Let n, d, S and S ′ be as given in Example 2.1. Consider q = 3 and therefore N = nq = 48. If D = {4, 12} then SN (D) = S48 (4) ∪ S48 (12) = {4, 20, 28, 44} ∪ {12, 36} = {4, 12, 20, 28, 36, 44}. Also we have qS = {3, 6, 9, 15, 33, 39, 42, 45} and qS ′ = {3, 6, 9, 39, 42, 45}. Notice that both sets qS ∩ SN (D) and qS ′ ∩ SN (D) are empty sets. Since |S ∩ Sn (d)| ≡ 2 (mod ⏐ 4), by ⏐Corollary 2.5, we conclude that( Cay (ZN , qS ∪ S)N (D)) admits PGST with respect to a sequence in 2π Z. Similarly ⏐S ′ ∩ Sn (d)⏐ ≡ 0 (mod 4) implies that Cay ZN , qS ′ ∪ SN (D) is almost periodic with respect to a sequence in 2π Z. Next we show that there are circulant graphs that fail to exhibit PGST. In Lemma 11 of [15], we observe that if m ∈ N and n = mp, where p is an odd prime, then the cycle Cay (Zn , {1, n − 1}) does not exhibit PGST. Now we show that Cay (Zn , S ) does not admit PGST if p ∤ s for all s ∈ S. Hence, Lemma 11 of [15], becomes a special case of the following Theorem. We use some methods used in [10] to establish the result.
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H. Pal / Discrete Mathematics 341 (2018) 889–895
Theorem 2.6. Let m ∈ N and n = mp, where p is an odd prime. Then the circulant graph Cay (Zn , S ) does not exhibit pretty good state transfer if p ∤ s for all s ∈ S. Proof. If m is an odd number then it is clear from Lemma 2.1 that the result follows. Thus, it is enough to prove the result for m even. For an odd prime p, let ωp be the primitive pth root of unity. Then for s ∈ S, since p ∤ s, we have 1 + ωps + ωp2s + · · · + ωp(p−1)s = 0. This further yields p−1
1+2
2 ∑
( cos
2rsπ
)
= 0.
p
r =1
(13)
Multiplying both sides of (13) by 2 cos p−1 2
λs +
∑
n
we obtain the following relation of eigenvalues of Cn (as given in (1)).
p−1 2
λs(mr +1) +
r =1
∑
λs(mr −1) = 0, s ∈ S .
(14)
r =1
Similarly multiplying (13) by 2 cos p−1 2
λ2s +
( 2sπ )
( 4sπ ) n
gives
p−1 2
∑
λs(mr +2) +
r =1
∑
λs(mr −2) = 0, s ∈ S .
(15)
r =1
Note that if λ′l is an eigenvalue of Cay (Zn , S ) then λ′l =
1 2
∑
s∈S
λsl . Hence, by Eqs. (14) and (15), we obtain
p−1
p−1
2 2 ( ′ ) ∑ ( ′ ) ( ′ ) ∑ λmr +2 − λ′mr +1 + λmr −2 − λ′mr −1 = 0. λ2 − λ′1 +
r =1
(16)
r =1
If Cay (Zn , S ) admits PGST then, by Eq. (2), we have a sequence of real numbers {tk } and a complex number γ with |γ | = 1 such that lim
k→∞
n−1 ∑
exp −i λ′l tk + lπ
[
(
)]
= nγ .
l=0
{ }
Since the unit circle is compact, we have a subsequence tk′ of {tk } such that lim exp[−i λ′l tk′ + lπ ] = γ , for 0 ≤ l ≤ n − 1.
(
)
k→∞
Moreover, this gives lim exp[−i λ′l+1 − λ′l tk′ ] = −1.
(
)
k→∞
Denoting the term in left hand side of Eq. (16) as L, it is evident that lim exp −iLtk′ = −1,
(
)
k→∞
but this is absurd as L = 0. Hence the circulant graph Cay (Zn , S ) does not exhibit PGST if p ∤ s for all s ∈ S. □ 3. Conclusions We have considered PGST on circulant graphs. In [15], the authors have classified all cycles and their compliments admitting PGST. There it is established that a cycle Cn exhibits PGST if and only if n is a power of 2. In this article, we have found several more circulant graphs on n vertices exhibiting PGST whenever n is a power of 2. Observe that, if n has an odd prime factor then there is no cycles on n vertices exhibiting PGST. However, in that case, we have found that there are other circulant graphs admitting PGST. Also we have found a large number of circulant graphs that do not admit PGST. The results established in this article generalize some of those results appearing in [15]. However, more work is required to classify all circulant graphs exhibiting PGST. Acknowledgments I would like to convey my deepest gratitude to the reviewers for carefully reading the manuscript. The reviewers’ valuable comments and suggestions helped me to improve the quality of the manuscript.
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E. Ackelsberg, Z. Brehm, A. Chan, J. Mundinger, C. Tamon, Laplacian state transfer in coronas, Linear Algebra Appl. 506 (2016) 154–167. E. Ackelsberg, Z. Brehm, A. Chan, J. Mundinger, C. Tamon, Quantum State Transfer in Coronas. arXiv:1605.05260. M. Bašić, Characterization of circulant networks having perfect state transfer, Quantum Inf. Process. 12 (1) (2013) 345–364. C.H. Bennett, G. Brassard, Quantum cryptography: public key distribution and coin tossing, in: Proc. IEEE Int. Conf. Computers Systems and Signal Processing, Bangalore, India, 1984, pp. 175-179. G. Coutinho, K. Guo, C.M. van Bommel, Pretty Good State Transfer Between Internal Nodes of Paths. arXiv:1611.09836. A.K. Ekert, Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett. 67 (6) (1991) 661. X. Fan, C. Godsil, Pretty good state transfer on double stars, Linear Algebra Appl. 438 (5) (2013) 2346–2358. C. Godsil, Periodic graphs, Electron. J. Combin. 18 (1) (2011) #P23. C. Godsil, State transfer on graphs, Discrete Math. 312 (1) (2012) 129–147. C. Godsil, S. Kirkland, S. Severini, J. Smith, Number-theoretic nature of communication in quantum spin systems, Phys. Rev. Lett. 109 (5) (2012) 050502. W. Klotz, T. Sander, Integral Cayley graphs over Abelian groups, Electron. J. Combin. 17 (2010) #R81. H. Pal, State Transfer on NEPS and Cayley Graphs (Ph.D. dissertation), Indian Institute of Technology, Guwahati, 2016. http://gyan.iitg.ernet.in/handle/ 123456789/848. H. Pal, B. Bhattacharjya, A class of gcd-graphs having perfect state transfer, Electron. Notes Discrete Math. 53 (2016) 319–329. H. Pal, B. Bhattacharjya, Perfect state transfer on NEPS of the path on three vertices, Discrete Math. 339 (2) (2016) 831–838. H. Pal, B. Bhattacharjya, Pretty good state transfer on circulant graphs, Electron. J. Combin. 24 (2) (2017) #P2.23. H. Pal, B. Bhattacharjya, Pretty good state transfer on some NEPS, Discrete Math. 340 (4) (2017) 746–752. H. Pal, B. Bhattacharjya, Perfect State Transfer on gcd-graphs, Linear Multilinear Algebra 65 (11) (2017) 2245–2256. N. Saxena, S. Severini, I.E. Shparlinski, Parameters of integral circulant graphs and periodic quantum dynamics, Int. J. Quantum Inf. 05 (2007) 417. W. So, Integral circulant graphs, Discrete Math. 306 (1) (2006) 153–158. C.M. van Bommel, A Complete Characterization of Pretty Good State Transfer on Paths. arXiv:1612.05603.
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
More circulant graphs exhibiting pretty good state transfer Hiranmoy Pal Department of Mathematics, Indian Institute of Information Technology Bhagalpur, Bihar, 813210, India
article
info
Article history: Received 27 May 2017 Received in revised form 2 November 2017 Accepted 18 December 2017 Available online 17 January 2018 Keywords: Graph Circulant graph Pretty good state transfer
a b s t r a c t The transition matrix of a graph G corresponding to the adjacency matrix A is defined by H(t) := exp (−itA) , where t ∈ R. The graph is said to exhibit pretty good state transfer between a pair of vertices u and v if there exists a sequence {tk } of real numbers such that limk→∞ H(tk )eu = γ ev , where γ is a complex number of unit modulus. We present a class of circulant graphs admitting pretty good state transfer. Also we find some circulant graphs not exhibiting pretty good state transfer. This generalizes several pre-existing results on circulant graphs admitting pretty good state transfer. © 2017 Elsevier B.V. All rights reserved.
1. Introduction In [9], Godsil first introduced the notion of pretty good state transfer (PGST) in the context of continuous-time quantum walks. State transfer has significant applications in quantum information processing and cryptography (see [4,6]). We consider state transfer with respect to the adjacency matrix of a graph. It can be considered with respect the Laplacian matrix as well. However, for a regular graph G, one can observe that G admits PGST with respect to the adjacency matrix if and only if G admits PGST with respect to the Laplacian matrix. All graphs we consider are simple, undirected and finite. Now we define PGST on a graph G with adjacency matrix A. The transition matrix of G is defined by H(t) := exp (−itA) , where t ∈ R. Let eu denote the characteristic vector corresponding to the vertex u of G. The graph G is said to exhibit PGST between a pair of vertices u and v if there is a sequence {tk } of real numbers such that lim H(tk )eu = γ ev , where γ ∈ C and |γ | = 1.
k→∞
In the past decade the study of state transfer has received considerable attention. Now we know a few graphs exhibiting PGST. The main goal here is to find new graphs that admits PGST. However it is more desirable to find PGST in graphs with large diameters. In [10], Godsil et al. established that the path on n vertices, denoted by Pn , exhibits PGST between the end vertices if and only if n + 1 equals to either 2m or p or 2p, where p is an odd prime. Further investigations have been done in [5] on the existence of PGST in internal vertices of some paths Pn . In [20], van Bommel established a complete characterization of PGST in Pn between any pair of vertices. A complete characterization of cycles exhibiting PGST appears in [15]. There we see that a cycle on n vertices exhibits PGST if and only if n is a power of 2. In [7], Fan and Godsil showed that a double star Sk,k admits PGST if and only if 4k + 1 is not a perfect square. A NEPS is a graph product which generalizes several well known graph products, namely, Cartesian product, Kronecker product etc. In [16], we see some NEPS of the path on three vertices having PGST. Also, in [1,2], PGST has been studied on corona products. Some other relevant results regarding state transfer can be found in [9,13,17,14,18]. In the present article, we find a class of circulant graphs admitting PGST. Also we find some E-mail address: [email protected]. https://doi.org/10.1016/j.disc.2017.12.012 0012-365X/© 2017 Elsevier B.V. All rights reserved.
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H. Pal / Discrete Mathematics 341 (2018) 889–895
circulant graphs not exhibiting pretty good state transfer. The results established in this article generalize several results appearing in [15] on circulant graphs admitting PGST. Let (Γ , +) be a finite abelian group and consider S \ {0} ⊆ Γ with {−s : s ∈ S } = S. A Cayley graph over Γ with the symmetric set S is denoted by Cay (Γ , S ). The graph has the vertex set Γ where two vertices a, b ∈ Γ are adjacent if and only if a − b ∈ S. The set S is called the connection set of Cay (Γ , S ). Let Zn be the cyclic group of order n. A circulant graph is a Cayley graph over Zn . The cycle Cn , in particular, is the circulant graph over Zn with the connection set {1, n − 1}. The eigenvalues and its corresponding eigenvectors of a cycle are very well known. The eigenvalues of Cn are
λl = 2 cos
(
2lπ
)
n
, 0 ≤ l ≤ n − 1,
(1)
[
l(n−1)
and the corresponding eigenvectors are vl = 1, ωnl , . . . , ωn
]T
, where ωn = exp
( 2π i ) n
is the primitive nth root of unity.
In general, the eigenvalues and eigenvectors of a Cayley graph over an abelian group are also known in terms of characters of the abelian group. In [11], it appears that the eigenvectors of a Cayley graph over an abelian group are independent of the connection set. Therefore the sets of eigenvectors of two Cayley graphs defined over an abelian group can be chosen to be equal. Hence we have the following result. Proposition 1.1. If S1 and S2 are symmetric subsets of an abelian group Γ then adjacency matrices of the Cayley graphs Cay(Γ , S1 ) and Cay(Γ , S2 ) commute. Let G and H be two simple graphs on the same vertex set V , and the respective edge sets E(G) and E(H). The edge union of G and H, denoted G ∪ H, is defined on the vertex set V whose edge set is E(G) ∪ E(H). Using Proposition 1.1 and the exponential nature of the transition matrix, we obtain the following result which allows us to find the transition matrix of an edge union of two edge disjoint Cayley graphs. Proposition 1.2 ([17]). Let Γ be a finite abelian group and consider two disjoint and symmetric subsets S , T ⊂ Γ . Suppose the transition matrices of Cay(Γ , S) and Cay(Γ , T ) are HS (t) and HT (t), respectively. Then Cay(Γ , S ∪ T ) has the transition matrix HS (t)HT (t). Let n ∈ N and d be a proper divisor of n. We denote Sn (d) = {x ∈ Zn : gcd(x, n) = d} , and for any set D of proper divisors of n, we define Sn (D) =
⋃
Sn (d).
d∈D
The set Sn (D) is called a gcd-set of Zn . A graph is called integral if all its eigenvalues are integers. The following theorem determines which circulant graphs are integral. Theorem 1.3 ([19]). A circulant graph Cay (Zn , S ) is integral if and only if S is a gcd-set. It is therefore clear that if a circulant graph Cay (Zn , S ) is non-integral then there exists a proper divisor d of n such that S ∩ Sn (d) is a non-empty proper subset of Sn (d). A graph G is said to exhibit perfect state transfer (PST) between a pair of vertices u and v if there exists t ∈ R such that H(t)eu = γ ev , where γ ∈ C and |γ | = 1. The graph G is said to be periodic at a vertex u if there is t(̸ = 0) ∈ R such that H(t)eu = γ eu , where γ ∈ C and |γ | = 1. Moreover, the graph G is said to be periodic if it is periodic at all vertices at the same time. More information regarding periodicity and PST can be found in [8]. It can be observed that PGST is a generalization to PST. However, in [12], we observe in Proposition 1.3.1 that if a graph is periodic then it admits PST if and only if it admits PGST. It is worth mentioning the fact presented in [18] that a circulant graph is periodic if and only if the graph is integral. In [3], Bašić considered integral circulant graphs and presented a complete characterization of integral circulant networks exhibiting PST. This motivates us to study non-integral circulant graphs for PGST. 2. Pretty good state transfer on circulant graphs Let us recall some important observations on circulant graphs exhibiting PGST (see [15]). We include the following discussion for convenience. Suppose G is a graph with adjacency matrix A. It is well known that the matrix P of an automorphism of G commutes with A. By the spectral decomposition of the transition matrix H(t) of G, we conclude that H(t) is a polynomial in A. Therefore P commutes with H(t) as well. Suppose G admits PGST between two vertices u and v . Then there exists a sequence {tk } of real numbers and γ ∈ C with |γ | = 1 such that lim H(tk )eu = γ ev .
k→∞
H. Pal / Discrete Mathematics 341 (2018) 889–895
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Further this implies that lim H(tk ) (Peu ) = γ (Pev ) .
k→∞
The sequence {H(tk )eu } cannot have two different limits, and therefore if P fixes eu then P must also fix ev . For the circulant graph Cay (Zn , S ), we know that the map sending j to n − j is an automorphism. As a consequence, we have the following result. Lemma 2.1 ([15]). If pretty good state transfer occurs in a circulant graph Cay (Zn , S ) then n is even and it occurs only between the pair of vertices u and u + 2n , where u, u + 2n ∈ Zn . A graph G is called vertex transitive if for any pair of vertices u, v in G there exists an automorphism sending u to v . All circulant graphs are well known to be vertex transitive. Thus, by Lemma 2.1, it is enough to find PGST in a circulant graph between the pair of vertices 0 and 2n . First we calculate the (0, 2n )th entry of the transition matrix of the cycle Cn . ∑n−1 1 ∗ Suppose A = l=0 λl El is the spectral decomposition of the adjacency matrix of Cn . Here notice that El = n vl vl , where
[
l(n−1)
vl = 1, ωnl , . . . , ωn
]T
as given in (1). Now we obtain the transition matrix of Cn as
H(t) = exp (−itA) =
n−1 ∑
exp (−iλl t ) El .
l=0
− nl 2
Since (0, 2n )th entry of El is 1n ωn H(t)0, n = 2
n−1 1∑
n
, we find the (0, 2n )th entry of H(t) as − nl 2
exp (−iλl t ) · ωn
=
l=0
n−1 1∑
n
exp [−i (λl t + lπ)] .
(2)
l=0
The next result finds some cycles admitting PGST with respect to a sequence in 2π Z. Later we extend this result to more general circulant graphs. Lemma 2.2 ([15]). A cycle Cn exhibits pretty good state transfer if n = 2k , k ≥ 2, with respect to a sequence in 2π Z. A graph G is said to be almost periodic if there is a sequence {tk } of real numbers and a complex number γ of unit modulus such that limk→∞ H(tk ) = γ I , where I is the identity matrix of appropriate order. Since the circulant graphs are vertex transitive, we observe that a circulant graph is almost periodic if and only if limk→∞ H(tk )e0 = γ e0 . This implies that if a cycle admits PGST then it is necessarily almost periodic. Hence, by Lemma 2.2, all cycles of size n = 2k , k ≥ 2, are almost periodic. However, in the next result, we find a connection between the time sequences of two cycles one of which exhibits PGST and the other one is almost periodic. Lemma 2.3. Let k, k′ ∈ N and 2 ≤ k′ < k. If the cycle C2k admits pretty good state transfer with respect to a sequence {tm } then C2k′ is almost periodic with respect to a sub-sequence of {tm }. Proof. We begin with the observation that if k′ < k then
( cos
2lπ
⎛ ( = cos ⎝
′
2k
′
2 l · 2k−k
)
2k
) ⎞ π ⎠.
Consequently, if θl ’s and λl ’s are the eigenvalues of C2k′ and C2k , respectively, then we necessarily have θl = λl·2k−k′ , where ′ 0 ≤ l ≤ 2k − 1. Suppose C2k admits PGST with respect to a time sequence {tm }. Therefore, by Eq. (2), there exists a complex number γ with |γ | = 1 such that 2k −1
lim
∑
m→∞
exp [−i (λl tm + lπ )] = 2k γ .
l=0
{ }
′ Since the unit circle is compact, we have a subsequence tm of {tm } such that ′ lim exp[−i λl tm + lπ ] = γ , for 0 ≤ l ≤ 2k − 1.
(
)
m→∞
This further gives ′
′ lim exp[−i θl tm ] = γ , for 0 ≤ l ≤ 2k − 1.
(
m→∞
)
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H. Pal / Discrete Mathematics 341 (2018) 889–895
Hence we have ′ 2k −1
lim
∑
m→∞
′ exp −i θl tm
[
(
)]
′
= 2k γ .
(3)
l=0
Notice that the (0, 0)th entry of the transition matrix of C2k′ is
{′}
1 ′ 2k
∑2k′ −1 l=0
exp [−i (θl t )] , where t ∈ R. Therefore, by Eq. (3),
we conclude that C2k′ is almost periodic with respect to tm . Hence we have the desired result. □ Now we extend Lemma 2.2 to obtain more circulant graphs (apart from the cycles) on 2k vertices exhibiting PGST. Also we find some circulant graphs which are almost periodic. It can well be observed that Theorem 7 appearing in [15] becomes a special case to the following theorem. Theorem 2.4. Let k ∈ N and n = 2k . Also let Cay (Zn , S ) be a non-integral circulant graph. Let d be the least among all the divisors of n so that S ∩ Sn (d) is a non-empty proper subset of Sn (d). If |S ∩ Sn (d)| ≡ 2 (mod 4) then Cay (Zn , S ) admits pretty good state transfer with respect to a sequence in 2π Z. Moreover, if |S ∩ Sn (d)| ≡ 0 (mod 4) then Cay (Zn , S ) is almost periodic with respect to a sequence in 2π Z. Proof. ( For each x ∈)S ∩ Sn (d), observe that Cay (Zn , {x, n − x}) is isomorphic to the disjoint( union of d copies ) of the cycle Cay Z n , 1,
} { } − 1 . By Lemma 2.2, there exists a sequence {tm } in 2π Z such that Cay Z nd , 1, nd − 1 admits PGST ( ( ) ) nx n with respect to {tm }. In particular, the cycle 0, x, . . . , 2d , . . . , nd − 1 x admits PGST between the pair of vertices 0 and 2d nx nx n between the pair of vertices 0 and 2d . Note that 2d ≡ 2 (mod n). This in turn implies that for all x ∈ S ∩ Sn (d), the graph Cay (Zn , {x, n − x}) exhibits PGST between the pair of vertices 0 and 2n with respect to {tm }. If HS ∩Sn (d) (t) is the transition matrix of Cay (Zn , S ∩ Sn (d)) then by Proposition 1.2, we can express HS ∩Sn (d) (t) as ∏ Hx (t), (4) HS ∩Sn (d) (t) = {
d
n d
x∈S ∩Sn (d), x< 2n
where Hx (t) denotes the transition matrix of Cay (Zn , {x, n − x}) for all x ∈ S ∩ Sn (d). In Eq. (4), notice that if |S ∩ Sn (d)| ≡ 2 (mod 4) then there are odd number of matrices in the product. Also we have just concluded that for all x ∈ S ∩ Sn (d) lim Hx (tm )e0 = γ e n , |γ | = 1.
(5)
2
m→∞
Since transition matrices are symmetric, we conclude that lim Hx (tm )e n = γ e0 , ∀x ∈ S ∩ Sn (d).
(6)
2
m→∞
After successive application of (5) and (6), we obtain the following from Eq. (4).
⎡ lim HS ∩Sn (d) (tm )e0 = lim ⎣
m→∞
m→∞
⎤ ∏
Hx (tm )⎦ e0 = γ
|S ∩Sn (d)| 2
en . 2
(7)
x∈S ∩Sn (d), x< 2n
′ Let S ′ = x ∈ S | gcd(x, n) = d′ and ( Sd′)⊂ S . Notice that if S is non-empty then it is in fact a gcd-set. Therefore, ( Theorem ) 1.3 implies that the sub-graph Cay Zn , S ′ of Cay (Zn , S ) is integral. Suppose HS ′ (t) is the transition matrix of Cay Zn , S ′ . Using spectral decomposition of HS ′ (t), we deduce that
{
}
HS ′ (t) = I , t ∈ 2π Z.
(8)
) Now consider S = S \ S ∪ (S ∩ Sn (d)) . By the assumption on the divisor d of n and the choice of S ′′ , we must have ′′ gcd(s, n) > d for all s ∈ S . This in turn implies that the )size of each disjoint appearing in Cay (Zn , {s, n − s}) is ( ( cycles { } { }) strictly less than the size of the cycle Cay Z n , 1, nd − 1 . Recall that Cay Z n , 1, nd − 1 admits PGST with respect d d to the sequence {tm }. Therefore, by Lemma 2.3, we have a subsequence of {tm } with respect to which each component ′′ of Cay (Zn , {s, n − s}) is almost periodic. { } Hence Cay (Zn , {s, n − s}) is ′′also almost periodic. Since S is finite, we can { } { }) appropriately choose a subsequence t of t such that for all s ∈ S , the graph Cay Z , s , n − s is{ almost ( mr m n { } ( ) } periodic ′′ with respect to tmr . Hence, by Proposition 1.2, the graph Cay Z , S is almost periodic with respect to t n mr . Therefore, ( ) if HS ′′ (t) is the transition matrix of Cay Zn , S ′′ then we have ( ) lim HS ′′ tmr e0 = ζ e0 , |ζ | = 1. (9) r →∞ { } { } Since tmr is a subsequence of {tm }, notice that Cay (Zn , S ∩ Sn (d)) also admits PGST with respect to tmr . Therefore, from (
′′
′
(7), we get lim HS ∩Sn (d) tmr e0 = γ
(
r →∞
)
|S ∩Sn (d)| 2
en . 2
(10)
H. Pal / Discrete Mathematics 341 (2018) 889–895
893
Note that S ∩ Sn (d), S ′ and S ′′ are disjoint subsets of S, and S = (S ∩ Sn (d)) ∪ S ′ ∪ S ′′ . Therefore, if H(t) is the transition matrix of Cay (Zn , S ) then by Proposition 1.2, using (8), (9) and (10), we obtain
(
)
[
(
)
(
)
(
lim H tmr e0 = lim HS ∩Sn (d) tmr · HS ′′ tmr · HS ′ tmr
r →∞
)]
r →∞
e0
= lim HS ∩Sn (d) tmr · lim HS ′′ tmr e0 , as tmr ∈ 2π Z r →∞ ( ) r →∞ = lim HS ∩Sn (d) tmr · ζ e0
(
r →∞ |S ∩Sn (d)|
= ζγ
2
)
)
(
en . 2
This proves our claim that Cay (Zn , S ) admits PGST with respect to a sequence in 2π Z. For the second part of the proof, observe that, if we consider |S ∩ Sn (d)| ≡ 0 (mod 4) then we get the following equation instead of (7).
⎡
⎤ ∏
lim HS ∩Sn (d) (tm )e0 = lim ⎣
m→∞
m→∞
Hx (tm )⎦ e0 = γ
|S ∩Sn (d)| 2
e0 .
x∈S ∩Sn (d), x< 2n
Then imitating the rest of the proof of the first part, we obtain the desired result. □ We illustrate Theorem 2.4 by the following example. Example 2.1. Let n = 16 and consider Cay (Z16 , S ) where S = {1, 2, 3, 5, 11, 13, 14, 15}. Here S ∩ S16 (1) = {1, 3, 5, 11, 13, 15} ⊊ S16 (1) and S ∩ S16 (2) = {2, 14} ⊊ S16 (2). In this case, we have d = 1. Notice that |S ∩ S16 (1)| ≡ 2 (mod 4). Hence, by Theorem 2.4, that Cay (Z16 , S ) admits PGST with respect to a sequence in 2⏐π Z. Observe ⏐ that if we consider the ( we conclude ) graph Cay Z16 , S ′ with S ′ = {(1, 2, 3, )13, 14, 15}, then proceeding as above, we obtain ⏐S ′ ∩ S16 (1)⏐ ≡ 0 (mod 4). Hence, by Theorem 2.4, we find that Cay Z16 , S ′ is almost periodic with respect to a sequence in 2π Z. Now we find even more circulant graphs that is either almost periodic or admits PGST. Corollary 2.5. Suppose all the conditions of Theorem 2.4 are satisfied. Let q ∈ 2N + 1, N = nq and D be any set of proper divisors of N such that qS ∩ SN (D) = ∅. If |S ∩ Sn (d)| ≡ 2 (mod 4) then Cay (ZN , qS ∪ SN (D)) admits PGST with respect to a sequence in 2π Z. Moreover, if |S ∩ Sn (d)| ≡ 0 (mod 4) then Cay (ZN , qS ∪ SN (D)) is almost periodic with respect to a sequence in 2π Z. Proof. Note that SN (D) is a gcd-set of ZN . Therefore, by Theorem 1.3, the graph Cay (ZN , SN (D)) is integral. If HSN (D) (t) be the transition matrix of Cay (ZN , SN (D)) then using spectral decomposition of HSN (D) (t), we deduce that HSN (D) (t) = I , t ∈ 2π Z.
(11)
Let H matrix of Cay (ZN , qS ). Since qS ∩ SN (D) = ∅, by Proposition 1.2, we find the transition matrix of ( qS (t) be the transition ) Cay ZN , qS ∪ Sq (D) as H(t) = HqS (t) · HSN (D) (t), t ∈ R. If t ∈ 2π Z then using (11), we get H(t) = HqS (t).
(12)
It can be observed that Cay (ZN , qS ) is isomorphic to q disjoint copies of Cay (Zn , S ). By Theorem 2.4, if |S ∩ Sn (d)| ≡ 2 (mod 4) then Cay (Zn , S ) admits PGST with respect to a sequence in 2π Z. Hence Cay (ZN , qS ) admits PGST with respect to a sequence in 2π Z as well. Similarly, if |S ∩ Sn (d)| ≡ 0 (mod 4) then Cay (ZN , qS ) is almost periodic with respect to a sequence in 2π Z. Finally, by (12), we have the desired result. □ We present the following example supporting Corollary 2.5. Example 2.2. Let n, d, S and S ′ be as given in Example 2.1. Consider q = 3 and therefore N = nq = 48. If D = {4, 12} then SN (D) = S48 (4) ∪ S48 (12) = {4, 20, 28, 44} ∪ {12, 36} = {4, 12, 20, 28, 36, 44}. Also we have qS = {3, 6, 9, 15, 33, 39, 42, 45} and qS ′ = {3, 6, 9, 39, 42, 45}. Notice that both sets qS ∩ SN (D) and qS ′ ∩ SN (D) are empty sets. Since |S ∩ Sn (d)| ≡ 2 (mod ⏐ 4), by ⏐Corollary 2.5, we conclude that( Cay (ZN , qS ∪ S)N (D)) admits PGST with respect to a sequence in 2π Z. Similarly ⏐S ′ ∩ Sn (d)⏐ ≡ 0 (mod 4) implies that Cay ZN , qS ′ ∪ SN (D) is almost periodic with respect to a sequence in 2π Z. Next we show that there are circulant graphs that fail to exhibit PGST. In Lemma 11 of [15], we observe that if m ∈ N and n = mp, where p is an odd prime, then the cycle Cay (Zn , {1, n − 1}) does not exhibit PGST. Now we show that Cay (Zn , S ) does not admit PGST if p ∤ s for all s ∈ S. Hence, Lemma 11 of [15], becomes a special case of the following Theorem. We use some methods used in [10] to establish the result.
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H. Pal / Discrete Mathematics 341 (2018) 889–895
Theorem 2.6. Let m ∈ N and n = mp, where p is an odd prime. Then the circulant graph Cay (Zn , S ) does not exhibit pretty good state transfer if p ∤ s for all s ∈ S. Proof. If m is an odd number then it is clear from Lemma 2.1 that the result follows. Thus, it is enough to prove the result for m even. For an odd prime p, let ωp be the primitive pth root of unity. Then for s ∈ S, since p ∤ s, we have 1 + ωps + ωp2s + · · · + ωp(p−1)s = 0. This further yields p−1
1+2
2 ∑
( cos
2rsπ
)
= 0.
p
r =1
(13)
Multiplying both sides of (13) by 2 cos p−1 2
λs +
∑
n
we obtain the following relation of eigenvalues of Cn (as given in (1)).
p−1 2
λs(mr +1) +
r =1
∑
λs(mr −1) = 0, s ∈ S .
(14)
r =1
Similarly multiplying (13) by 2 cos p−1 2
λ2s +
( 2sπ )
( 4sπ ) n
gives
p−1 2
∑
λs(mr +2) +
r =1
∑
λs(mr −2) = 0, s ∈ S .
(15)
r =1
Note that if λ′l is an eigenvalue of Cay (Zn , S ) then λ′l =
1 2
∑
s∈S
λsl . Hence, by Eqs. (14) and (15), we obtain
p−1
p−1
2 2 ( ′ ) ∑ ( ′ ) ( ′ ) ∑ λmr +2 − λ′mr +1 + λmr −2 − λ′mr −1 = 0. λ2 − λ′1 +
r =1
(16)
r =1
If Cay (Zn , S ) admits PGST then, by Eq. (2), we have a sequence of real numbers {tk } and a complex number γ with |γ | = 1 such that lim
k→∞
n−1 ∑
exp −i λ′l tk + lπ
[
(
)]
= nγ .
l=0
{ }
Since the unit circle is compact, we have a subsequence tk′ of {tk } such that lim exp[−i λ′l tk′ + lπ ] = γ , for 0 ≤ l ≤ n − 1.
(
)
k→∞
Moreover, this gives lim exp[−i λ′l+1 − λ′l tk′ ] = −1.
(
)
k→∞
Denoting the term in left hand side of Eq. (16) as L, it is evident that lim exp −iLtk′ = −1,
(
)
k→∞
but this is absurd as L = 0. Hence the circulant graph Cay (Zn , S ) does not exhibit PGST if p ∤ s for all s ∈ S. □ 3. Conclusions We have considered PGST on circulant graphs. In [15], the authors have classified all cycles and their compliments admitting PGST. There it is established that a cycle Cn exhibits PGST if and only if n is a power of 2. In this article, we have found several more circulant graphs on n vertices exhibiting PGST whenever n is a power of 2. Observe that, if n has an odd prime factor then there is no cycles on n vertices exhibiting PGST. However, in that case, we have found that there are other circulant graphs admitting PGST. Also we have found a large number of circulant graphs that do not admit PGST. The results established in this article generalize some of those results appearing in [15]. However, more work is required to classify all circulant graphs exhibiting PGST. Acknowledgments I would like to convey my deepest gratitude to the reviewers for carefully reading the manuscript. The reviewers’ valuable comments and suggestions helped me to improve the quality of the manuscript.
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