Pretty good state transfer on 1-sum of star graphs

Hailong Hou; Rui Gu; Mengdi Tong

doi:10.1515/math-2018-0119

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Pretty good state transfer on 1-sum of star graphs

Hailong Hou , Rui Gu and Mengdi Tong

Published/Copyright: December 27, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

Let A be the adjacency matrix of a graph G and suppose U(t) = exp(itA). We say that we have perfect state transfer in G from the vertex u to the vertex v at time t if there is a scalar γ of unit modulus such that U(t)e_u = γ e_v. It is known that perfect state transfer is rare. So C.Godsil gave a relaxation of this definition: we say that we have pretty good state transfer from u to v if there exists a complex number γ of unit modulus and, for each positive real ϵ there is a time t such that ‖U(t)e_u–γ e_v‖ < ϵ. In this paper, the quantum state transfer on 1-sum of star graphs F_k,l is explored. We show that there is no perfect state transfer on F_k,l, but there is pretty good state transfer on F_k,l if and only if k = l.

Keywords: Perfect state transfer; Pretty good state transfer; Kronecker approximation theorem

MSC 2010: 05C50

1 Introduction

Quantum walks on graphs are a natural generalization of classical random walks. In recent years much attention has been paid to quantum walks of graphs and many interesting results concerning graphs and their continuous quantum walks have been obtained (see [1, 2, 5, 11,12,13] and references therein). It has been applied to key distributions in commercial cryptosystems, and it seems likely that further applications will be found.

Let G be a graph with adjacency matrix A and let U(t) denote the matrix-valued function exp(itA). It is known that U(t) is both symmetric and unitary, and that it determines a continuous quantum walk on G. If u ∈ V(X)), we use e_u to denote the standard basis vector indexed by u. If u and v are distinct vertices in G, we say that we have perfect state transfer from u to v at time t if there exists a complex number γ of unit modulus such that

U(t)eu=γev.

There is considerable literature on perfect state transfer. In [5, 6], Christandl et al showed that there is perfect state transfer between the end vertices of the paths P₂ and P₃, but perfect state transfer does not occur on a path on four or more vertices. Some results on perfect state transfer in gcd-graph can be found in [14]. From [8], Godsil showed that, for any integer k, there are only finite many connected graphs with valency at most k on which perfect state transfer occurs.

Perfect state transfer is rare and we consider a relaxation of it. We say that we have pretty good state transfer from u to v if there exists a complex number γ of unit modulus and, for each positive real ϵ there is a time t such that

∥U(t)eu−γev∥<ϵ.

The notion of pretty good state transfer was first introduced by Godsil in [7], since this, there have been multiple published papers which have enhanced his work. In [11], Godsil et al showed that there is pretty good state transfer between the end-vertices on P_n if and only if n + 1 equals to 2^m or p or 2p, where p is an odd prime. Fan and Godsil [4] have studied pretty good state transfer on double star. They showed that a double star S_k,k admits pretty good state transfer if and only if 4k + 1 is not a perfect square. Pal [13] showed that a cycle C_n exhibits pretty good state transfer if and only if n = 2^k for some k ≥ 2.

Let X and Y be two graphs. We say Z is a 1-sum of X and Y at u if Z is obtained by merging the vertex u of X with the vertex u of Y and no edge joins a vertex of X∖ u with a vertex of Y∖ u. Denoted by F_k,l, the 1-sum of two star at a vertex u of degree 1. In this paper, the state transfer in quantum walk of these graphs is explored. We show that there is no perfect state transfer in these graphs, but there is pretty good state transfer when k = l.

2 Perfect State transfer

We would like to introduce a useful tool. For more expansive treatment, the refer is referred to [10].

Let G be a graph and π = {C₁, C₂,⋯, C_r} be a partition of V(X). We say π is equitable if every vertex in C_i has the same numbers b_ij of neighbors in C_j. The quotient graph of G induced by π, denoted by G/π, is a directed graph with vertex set π and b_ij arcs from the ith to jth cells of π. The entries of the adjacency matrix of G/π are given by A(G/π)_ij = b_ij. We can symmetrize A(G/π) to B by letting Bij=bijbji. We call the weighted graph with adjacency matrix B, the symmetrized quotient graph. In the following, we always use B to denote the symmetrized form of the matrix A(G/π). We list some known results which will be used within this paper.

Lemma 2.1

([3]). Let G be a graph and let u, v be two vertices of G. Then there is perfect state transfer between u and v at time t with phase γ if and only if all of the following conditions hold.

Vertices u and v are strongly cospectral.
There are integers a, △ where △ is square-free so that for each eigenvalue λ in supp_G(u):
1. λ=12(a+bλ△),for some integer b_λ.
2. euTEλ(G)evis positive if and only if(ρ(G)−λ)/g△is even, where
  g:=gcd({ρ(G)−λ△}:λ∈SuppG(u))

Moreover, if the above conditions hold, then the following also hold.

There is a minimum time of perfect state transfer between u and v given by
t0:=πg△
The time of perfect state transfer t is an odd multiple of t₀.
The phase of perfect state transfer is given by γ = e^–itρ(G).

Lemma 2.2

([8). Let G be a graph with perfect state transfer between vertices u and v. Then, for each automorphism τ ∈ Aut(G), τ(u) = u if and only if τ(v) = v.

Lemma 2.3

([10]). Suppose G has pretty good state transfer between vertices u and v. Then u and v are strongly cospectral, and each automorphism fixing u must fix v.

Lemma 2.4

([2]). Let X be a graph with an equitable π and assume {a} and {b} are singleton cells of π. Let B denote the adjacency matrix of the symmetrized quotient graph relative π. Then for any time t,

(e−itA(X))a,b=(e−itB){a},{b}

and therefore X has perfect state transfer from a to b at time t if and only if the symmetrized quotient graph has perfect state transfer from {a} to {b}.

Let S_k+1 and S_l+1 be the two star graphs with k + 1 and l + 1 edges respectively and w be a vertex of degree one in S_k+1 and S_l+1. Then the 1-sum of S_k+1 and S_l+1, denoted by F_k,l, is obtained by merging the vertex w of S_k+1 with the vertex w of S_l+1 and no edge joins a vertex of S_k+1∖ w with a vertex of S_l+1∖ w.

Lemma 2.5

There is no perfect state transfer from u₁to u₂on F_2,l(Fig. 1).

$Fig. 1 Graph F2l$

Fig. 1

Graph F_2l

Proof

Let F_2,l be a graph shown in Fig. 1. Let π be the equitable partition with cells

{{u1},{u2},{u},{w},{v},N(v)∖{w}}.

Then the adjacency matrix B of the corresponding symmetrized quotient graph induced by π is

00100000100011010000101000010l0000l0.

The eigenvalue of B are

θ₁ = 0 with multiplicity 2,

θ2=22l+4+l2−4l+8,

θ3=−22l+4+l2−4l+8,

θ4=22l+4−l2−4l+8,

θ5=−22l+4−l2−4l+8.

If there is perfect state transfer from {u₁} to {u₂} on F_2,l/π, by Lemma 2.1,

θ2−θ3θ4−θ5

is rational. Therefore

(θ2−θ3)2(θ4−θ5)2=l+4+l2−4l+8l+4−l2−4l+8=l2−2l+126l+4+(l+4)l2−4l+8)6l+4

is rational, and hence l²–4l+8 is a perfect square. This means that there exists a integer p such that l²–4l+8 = (l–2)² + 4 = p². Note that both l–2 and p are integers. This can occur only if l = 2. But in this case (θ₂–θ₃)/(θ₄–θ₅) = 2 is irrational. This is a contradiction. Hence there is no perfect state transfer from {u₁} to {u₂} on F_{2, l}/π and by Lemma 2.4, there is no perfect state transfer from u₁ to u₂ on F_2,l.□

Next we start to study perfect state transfer between two vertices u and v on F_k,k (see Fig. 2).

$Fig. 2 Graph Fk,k$

Fig. 2

Graph F_k,k

Lemma 2.6

There is no perfect state transfer from u to v on F_k,k.

Proof

Suppose F_k,k be a graph shown in Fig.2. Let π be the equitable partition with cells

{N(u)∖{w},{u},{w},{v},N(v)∖{w}}.

Then the adjacency matrix B of the corresponding symmetrized quotient graph induced by π is

0k000k0100010100010k000k0.

The eigenvalue of B are

θ1=0,θ2=k,θ3=−k,θ4=k+2,θ5=−k+2.

The corresponding eigenvectors are the columns of the following matrix:

111kk01−1k+2−k+2−k00220−11k+2−k+21−1−1kk.

It is easy to check that θ_i ∈ Supp_G({u}) for i = 2,3,4,5. If there is perfect state transfer from {u} to {v} on F_k,k/π, by Lemma 2.1, there exist integers m,n and a square-free integer p such that θ₂–θ₃ = mp and θ₄–θ₅ = np. Then n > m ≥ 1. Now

(n2−m2)p=n2p−m2p=(k+2)−k=2.

This is impossible. Hence there is no perfect state transfer from {u} to {v} on F_k,k/π and by Lemma 2.4, there is no perfect state transfer from u to v on F_k,k.□

Lemma 2.7

w is not strongly cospectral with the other vertices.

Proof

If w is strongly cospectral with the vertex a, then they must have the same degree. Thus a ≠ u_i (i = 1,2, ⋯, k) and a ≠ v_j (i = 1,2, ⋯, k). Without loss of generality, suppose a = u. Then k = 1. Note that F_1,l∖{u} = K₁∪ S_l+1 and F_1,l∖ \{w} = K₂∪ S_l. TThese are not cospectral, hence it follows they are not strongly cospectral which in itself provides a contradiction.□

Theorem 2.8

There is no perfect state transfer on F_k,l.

Proof

We divide into four cases to discuss:

Note that if there exists perfect state transfer from a to b, then a and b must be strongly cospectral. Since w is not strongly cospectral with the other vertices, there is no perfect state transfer from w to other vertices.
If there exists perfect state transfer from u to v, then u and v must have the same degree. This means that k = l. By Lemma 2.7, there is no perfect state transfer from u to v on F_k,k.
If there exists perfect state transfer from u_i to u_j, by Lemma 2.2, k = 2. By Lemma 2.5, there is no perfect state transfer from u₁ to u₂ on F_2,l. A similar argument will show that there is no perfect state transfer from v_i to v_j.
If there exists perfect state transfer from u_i to v_j, by Lemma 2.2, k = l = 1. Then the graph is P₅. We know that there is no perfect state transfer in P₅ from [5].□

3 Pretty Good State transfer

In this section, we will investigate pretty good state transfer on 1-sum of star graphs. We first introduce Kronecker approximation theorem on simultaneous approximation of numbers, which will be used later.

Lemma 3.1

([12]). Let 1, λ₁, ⋯, λ_mbe linearly independent over Q. Let α₁, ⋯, α_m be arbitrary real numbers, and let N, ϵ be positive real numbers. Then there are integers l > N and q₁, ⋯, q_m so that

|lλk−qk−αk|<ϵ,

for each k = 1,⋯, m.

Lemma 3.2

There is no pretty good state transfer from u₁to u₂on F_2,l.

Proof

Suppose θ_i (i = 1,2, ⋯,5) is the eigenvalue of B given in Lemma 2.6. Then the corresponding eigenvectors of θ₁ are

x11=(−1,1,0,0,0,0)T,x12=(l2,l2,0,−l,0,1)T.

Then θ₁ ∈ Supp({u₁}). Denote by E₁ the orthogonal projection onto the eigenvector belonging to θ₁ = 0. Note that x₁₁ and x₁₂ are orthogonal. Then

E1e{u1}=l2(1,−1,0,0,0,0)T+23l+2(l4,l4,0,−l,0,1)T,E1e{u2}=l2(−1,1,0,0,0,0)T+23l+2(l4,l4,0,−l,0,1)T.

Therefore E₁e_{u₁} ≠ ± E₁e_{u₂}. This means that {u₁} and {u₂} are not strongly cospectral. By Lemma 2.3, there is no pretty good state transfer between {u₁} and {u₂}. Therefore, by Lemma 2.4, there is no pretty good state transfer between u₁ and u₂.□

Lemma 3.3

There is pretty good state transfer from u to v on F_k,kfor any positive integer k.

Proof

Let π be the equitable partition with cells {N(u)∖ {w}, {u}, {w}, {v}, N(v)∖{w}} and B be the adjacency matrix of the corresponding symmetrized quotient graph induced by π. The eigenvalues and its corresponding eigenvectors are given in Lemma 2.6. Hence

exp(itB){u},{u}=∑r=15exp(itθr)(Er){u},{u} =−12cos⁡(tθ2)+12cos⁡(tθ4).

Hence there is pretty good state transfer from {u} to {v} if and only if there exists a sequence of times (t_i)_{i ≥ 0} such that

limi→∞cos⁡(tiθ2)=−limi→∞cos⁡(tiθ4)=±1.

In the following, we will show that there exists a sequence of times (t_i)_{i ≥ 0} such that lim_{i→ ∞} cos (t_iθ₂) = –lim_{i→ ∞} cos (t_iθ₄) = –1, which holds if there exist m,n ∈ Z such that t_iθ₂ ≈ (2m + 1)π and t_i θ₄≈ 2n π. The question becomes whether we can choose m,n such that

nkk+2−m≈12.

If kk+2 is rational, then there exist integers p,q and a square-free integer △ such that k=p△ and k+2=q△. Then q > p ≥ 1. Now

(q2−p2)△=q2△−p2△=(k+2)−k=2,

which is impossible. Hence kk+2 is irrational and so 1 and kk+2 are linearly independent. By Kronecker approximation theorem there exist m,n ∈ Z such that for any positive real number ϵ,

|nkk+2−m−12|<ϵ

This means that we can choose m,n such that nkk+2−m≈12 and so there exists a sequence of times (t_i)_{i ≥ 0} such that lim_{i→ ∞} cos (t_i θ₂) = –lim_{i→ ∞} cos (t_iθ₄) = –1.□

Theorem 3.4

There is pretty good state transfer on F_k,lif and only if

k = l and pretty good state transfer occurs between u and v;
k = l = 1 and pretty good state transfer occurs between its end vertices.

Proof

We divide into four cases to discuss, which is similar to Theorem 2.8:

By Lemma 2.7w is not strongly cospectral with the other vertices. By Lemma 2.3 there is no pretty good state transfer from w to other vertices.
If there exists pretty good state transfer from u to v, then u and v must have the same degree. This means that k = l. By Lemma 3.3, pretty good transfer occurs between u and v in this case.
If there exists pretty good state transfer from u_i to u_j, by Lemma 2.3, k = 2. But by Lemma 3.1, pretty good state transfer does not occur between u₁ and u₂ on F_2,l in this case. A similar argument will show that there is no pretty good state transfer between v_i and v_j.
If there exists perfect state transfer from u_i to v_j, by Lemma 2.3, k = l = 1. Thus the graph is P₅. We know that there is pretty good state transfer on P₅ between its end vertices.□

Acknowledgement

The authors want to express their gratitude to the referees for their helpful suggestions and comments. The first author Hailong Hou has been a visitor at the Department of Combinatorics and Optimization in University of Waterloo from November 2016 to November 2017. He wants to express his gratitude to professor Chris Godsil for his guidance and support.

This research was partially supported by the National Natural Science Foundation of China (No.11601337 and 11301151), the Key Project of the Education Department of Henan Province (No.13A110249) and the Innovation Team Funding of Henan University of Science and Technology (NO.2015XTD010).

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Received: 2018-04-28

Accepted: 2018-10-23

Published Online: 2018-12-27

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/math-2018-0119

Keywords for this article

Perfect state transfer; Pretty good state transfer; Kronecker approximation theorem

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