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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

# Pretty good state transfer on 1-sum of star graphs

Hailong Hou
• Corresponding author
• School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, Henan, 471023, China
• Email
• Other articles by this author:
/ Rui Gu
• School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, Henan, 471023, China
• Other articles by this author:
/ Mengdi Tong
• School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, Henan, 471023, China
• Other articles by this author:
Published Online: 2018-12-27 | DOI: https://doi.org/10.1515/math-2018-0119

## 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)eu = γ ev. 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)euγ ev‖ < ϵ. In this paper, the quantum state transfer on 1-sum of star graphs Fk,l is explored. We show that there is no perfect state transfer on Fk,l, but there is pretty good state transfer on Fk,l if and only if k = l.

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 uV(X)), we use eu 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 Pn if and only if n + 1 equals to 2m 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 Sk,k admits pretty good state transfer if and only if 4k + 1 is not a perfect square. Pal [13] showed that a cycle Cn exhibits pretty good state transfer if and only if n = 2k 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 Xu with a vertex of Yu. Denoted by Fk,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,⋯, Cr} be a partition of V(X). We say π is equitable if every vertex in Ci has the same numbers bij of neighbors in Cj. The quotient graph of G induced by π, denoted by G/π, is a directed graph with vertex set π and bij 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 = bij. We can symmetrize A(G/π) to B by letting $\begin{array}{}{B}_{ij}=\sqrt{{b}_{ij}{b}_{ji}}.\end{array}$ 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.

1. Vertices u and v are strongly cospectral.

2. There are integers a, △ where △ is square-free so that for each eigenvalue λ in suppG(u):

1. $\begin{array}{}\lambda =\frac{1}{2}\left(a+{b}_{\lambda }\sqrt{\mathrm{△}}\right),\end{array}$ for some integer bλ.

2. $\begin{array}{}{e}_{u}^{T}{E}_{\lambda }\left(G\right){e}_{v}\end{array}$ is positive if and only if $\begin{array}{}\left(\rho \left(G\right)-\lambda \right)/g\sqrt{\mathrm{△}}\end{array}$ is even, where

$g:=gcd({ρ(G)−λ△}:λ∈SuppG(u))$

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

1. There is a minimum time of perfect state transfer between u and v given by

$t0:=πg△$

2. The time of perfect state transfer t is an odd multiple of t0.

3. The phase of perfect state transfer is given by γ = eitρ(G).

#### Lemma 2.2

([8). Let G be a graph with perfect state transfer between vertices u and v. Then, for each automorphism τAut(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 Sk+1 and Sl+1 be the two star graphs with k + 1 and l + 1 edges respectively and w be a vertex of degree one in Sk+1 and Sl+1. Then the 1-sum of Sk+1 and Sl+1, denoted by Fk,l, is obtained by merging the vertex w of Sk+1 with the vertex w of Sl+1 and no edge joins a vertex of Sk+1w with a vertex of Sl+1w.

#### Lemma 2.5

There is no perfect state transfer from u1 to u2 on F2,l (Fig. 1).

Fig. 1

Graph F2l

#### Proof

Let F2,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,

$\begin{array}{}{\theta }_{2}=\frac{\sqrt{2}}{2}\sqrt{l+4+\sqrt{{l}^{2}-4l+8}},\end{array}$

$\begin{array}{}{\theta }_{3}=-\frac{\sqrt{2}}{2}\sqrt{l+4+\sqrt{{l}^{2}-4l+8}},\end{array}$

$\begin{array}{}{\theta }_{4}=\frac{\sqrt{2}}{2}\sqrt{l+4-\sqrt{{l}^{2}-4l+8}},\end{array}$

$\begin{array}{}{\theta }_{5}=-\frac{\sqrt{2}}{2}\sqrt{l+4-\sqrt{{l}^{2}-4l+8}}.\end{array}$

If there is perfect state transfer from {u1} to {u2} on F2,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 (θ2θ3)/(θ4θ5) = $\begin{array}{}\sqrt{2}\end{array}$ is irrational. This is a contradiction. Hence there is no perfect state transfer from {u1} to {u2} on F2, l/π and by Lemma 2.4, there is no perfect state transfer from u1 to u2 on F2,l.□

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

Fig. 2

Graph Fk,k

#### Lemma 2.6

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

#### Proof

Suppose Fk,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 θiSuppG({u}) for i = 2,3,4,5. If there is perfect state transfer from {u} to {v} on Fk,k/π, by Lemma 2.1, there exist integers m,n and a square-free integer p such that θ2θ3 = m $\begin{array}{}\sqrt{p}\end{array}$ and θ4θ5 = n $\begin{array}{}\sqrt{p}\end{array}$. 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 Fk,k/π and by Lemma 2.4, there is no perfect state transfer from u to v on Fk,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 aui (i = 1,2, ⋯, k) and avj (i = 1,2, ⋯, k). Without loss of generality, suppose a = u. Then k = 1. Note that F1,l∖{u} = K1Sl+1 and F1,l∖ \{w} = K2Sl. 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 Fk,l.

#### Proof

We divide into four cases to discuss:

1. 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.

2. 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 Fk,k.

3. If there exists perfect state transfer from ui to uj, by Lemma 2.2, k = 2. By Lemma 2.5, there is no perfect state transfer from u1 to u2 on F2,l. A similar argument will show that there is no perfect state transfer from vi to vj.

4. If there exists perfect state transfer from ui to vj, 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, ⋯, λm be linearly independent over Q. Let α1, ⋯, αm be arbitrary real numbers, and let N, ϵ be positive real numbers. Then there are integers l > N and q1, ⋯, qm so that

$|lλk−qk−αk|<ϵ,$

for each k = 1,⋯, m.

#### Lemma 3.2

There is no pretty good state transfer from u1 to u2 on F2,l.

#### Proof

Suppose θi (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 θ1Supp({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 E1 e{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 Fk,k for 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 (ti)i ≥ 0 such that

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

In the following, we will show that there exists a sequence of times (ti)i ≥ 0 such that limi→ ∞ cos (tiθ2) = –limi→ ∞ cos (tiθ4) = –1, which holds if there exist m,nZ such that tiθ2 ≈ (2m + 1)π and ti θ4≈ 2n π. The question becomes whether we can choose m,n such that

$nkk+2−m≈12.$

If $\begin{array}{}\sqrt{\frac{k}{k+2}}\end{array}$ is rational, then there exist integers p,q and a square-free integer △ such that $\begin{array}{}\sqrt{k}=p\sqrt{△}\end{array}$ and $\begin{array}{}\sqrt{k+2}=q\sqrt{△}.\end{array}$ Then q > p ≥ 1. Now

$(q2−p2)△=q2△−p2△=(k+2)−k=2,$

which is impossible. Hence $\begin{array}{}\sqrt{\frac{k}{k+2}}\end{array}$ is irrational and so 1 and $\begin{array}{}\sqrt{\frac{k}{k+2}}\end{array}$ are linearly independent. By Kronecker approximation theorem there exist m,nZ such that for any positive real number ϵ,

$|nkk+2−m−12|<ϵ$

This means that we can choose m,n such that n $\begin{array}{}\sqrt{\frac{k}{k+2}}-m\approx \frac{1}{2}\end{array}$ and so there exists a sequence of times (ti)i ≥ 0 such that limi→ ∞ cos (ti θ2) = –limi→ ∞ cos (tiθ4) = –1.□

#### Theorem 3.4

There is pretty good state transfer on F k,l if and only if

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

2. 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:

1. By Lemma 2.7 w is not strongly cospectral with the other vertices. By Lemma 2.3 there is no pretty good state transfer from w to other vertices.

2. 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.

3. If there exists pretty good state transfer from ui to uj, by Lemma 2.3, k = 2. But by Lemma 3.1, pretty good state transfer does not occur between u1 and u2 on F2,l in this case. A similar argument will show that there is no pretty good state transfer between vi and vj.

4. If there exists perfect state transfer from ui to vj, 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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Accepted: 2018-10-23

Published Online: 2018-12-27

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1483–1489, ISSN (Online) 2391-5455,

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