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State transfer and star complements in graphs
Discrete Applied Mathematics (
)
–
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State transfer and star complements in graphs Jiang Zhou a,b,∗ , Changjiang Bu b a
College of Computer Science and Technology, Harbin Engineering University, Harbin 150001, PR China
b
College of Science, Harbin Engineering University, Harbin 150001, PR China
article
abstract
info
Let G be a graph with adjacency matrix A, and let H (t ) = exp(itA). For an eigenvalue µ of A with multiplicity k, a star set for µ in G is a vertex set X of G such that |X | = k and the induced subgraph G − X does not have µ as an eigenvalue. G is said to have perfect state transfer from the vertex u to the vertex v if there is a time τ such that |H (τ )u,v | = 1. The unitary operator H (t ) has important applications in the transfer of quantum information. In this paper, we give an expression of H (t ). For a star set X of graph G, perfect state transfer does not occur between any two vertices in X . We also give some results for the existence of perfect state transfer in a graph. © 2013 Elsevier B.V. All rights reserved.
Article history: Received 12 June 2012 Received in revised form 24 April 2013 Accepted 19 August 2013 Available online xxxx Keywords: Adjacency matrix Perfect state transfer Star set Star complement Main eigenvalue
1. Introduction Let G be a simple undirected graph with adjacency matrix A. The eigenvalues of A are called the eigenvalues of G. Let H (t ) denote the matrix function H (t ) = exp(itA) =
√
∞ n n n i t A n=0
n!
,
where i = −1. Since A is symmetric, H (t ) is symmetric. Notice that the conjugate transpose of H (t ) is exp(−itA) = H (t )−1 . Hence H (t ) is a unitary matrix. We use Mu,v or (M )u,v to denote the (u, v)-entry of a matrix M. We say that G has perfect state transfer from vertex u to vertex v if there exists a time τ such that |H (τ )u,v | = 1 (see [7,8,16,17]). Some graphs with perfect state transfer are given in [2,1,3–5,7,14,15]. Let G be a graph with an eigenvalue µ of multiplicity k. A star set for µ in G is a vertex set X of G such that |X | = k and the induced subgraph G − X does not have µ as an eigenvalue. In this situation, G − X is called a star complement for µ in G. It is known that star sets exist for any eigenvalue of any graph (see [10,13,12]). The operator H (t ) = exp(itA) has important applications in the transfer of quantum information, it induces a continuoustime quantum walk (see [6,16]). In this paper, we give an expression of H (t ) = exp(itA). For a star set X of graph G, we show that perfect state transfer does not occur between any two vertices in X . We also give some results for the existence of perfect state transfer in a graph. 2. Preliminaries For a vertex u of graph G, we use eu to denote the unit vector whose u-th coordinate is 1 (i.e., eu is the standard basis vector with respect to u). Let A be the adjacency matrix of G. Suppose θ1 , . . . , θm are all distinct eigenvalues of A, and A has
∗
Corresponding author at: College of Computer Science and Technology, Harbin Engineering University, Harbin 150001, PR China. Tel.: +86 82519384. E-mail addresses: [email protected] (J. Zhou), [email protected] (C. Bu).
0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.08.028
2
J. Zhou, C. Bu / Discrete Applied Mathematics (
)
–
spectral decomposition A = r =1 θr Er , where Er is the orthogonal projection on the eigenspace associated with θr . Godsil shows that Er eu = ±Er ev (r = 1, . . . , m) if G perfect state transfer from vertex u to vertex v (cf. [8, Corollary 2.2]). For any eigenvector y of θr , we have y⊤ Er = y⊤ (r = 1, . . . , m). It is not difficult to obtain the following lemma.
m
Lemma 2.1. Let y be an eigenvector of the adjacency matrix of graph G. If G has perfect state transfer from u to v , then y⊤ eu = ±y⊤ ev . Lemma 2.2 ([13,12]). Let X be a set of k vertices in the graph G, and G has adjacency matrix
AX B
B⊤ C
, where AX is the adjacency
matrix of the subgraph induced by X . Then X is a star set for an eigenvalue µ of G if and only if µ is not an eigenvalue of C and
µI − AX = B⊤ (µI − C )−1 B.
z
In this situation, the eigenspace of µ consists of the vectors (µI − C )−1 Bz , where z ∈ Rk is an arbitrary vector. The multiset of eigenvalues of graph G is called the spectrum of G. Vertices u and v in G are called cospectral if G − u and G − v has the same spectrum. Lemma 2.3 ([8]). If graph G has perfect state transfer from u to v , then u and v are cospectral vertices in G. Let G be a graph with vertex set V . The partition V = V1 ∪ V2 ∪ · · · ∪ Vk is an equitable partition if each vertex in the cell Vi has the same number of neighbours in the cell Vj for all i, j ∈ {1, 2, . . . , k}. Lemma 2.4 ([8]). Suppose V1 and V2 are cells of an equitable partition π of graph G, and we have perfect state transfer from u in V1 to v in V2 at time τ . If y is the characteristic vector of the cell V1 , then H (τ )y is the scalar multiple of the characteristic vector of the cell V2 . Furthermore we have |V1 | = |V2 |. Lemma 2.5. Let G be a k-regular graph with adjacency matrix A, and µ is not an eigenvalue of G. Then (µI − A)−1 j = (µ− k)−1 j, where j is the all-one vector. Proof. Since G is k-regular, its largest eigenvalue is k, and the all-one vector j is the eigenvector of k. Then (µI − A)j = (µ − k)j, (µI − A)−1 j = (µ − k)−1 j. An eigenvalue µ of graph G is said to be a main eigenvalue of G if the eigenspace of µ is not orthogonal to the all-one vector (see [11]). It is not difficult to obtain the following lemma from [11, Proposition 2.7]. Lemma 2.6 ([11]). Let G be a graph with adjacency matrix M =
B⊤ C
A B
, where C is the adjacency matrix of a star complement
for an eigenvalue µ of G. Then µ is a non-main eigenvalue if and only if B⊤ (µI − C )−1 j = −j, where j is the all-one vector. We use Γ (v) to denote the set of neighbours of vertex v in a graph. A star set X of graph G is said to be uniform, if there exists a positive integer b such that |Γ (v) ∩ X | = b for every vertex v in the star complement G − X . Lemma 2.7 ([10]). Let X be a uniform star set for an eigenvalue µ of graph G, say |Γ (v) ∩ X | = b for all v in the star complement G − X . If µ is not a main eigenvalue of G, then the graph induced by X is regular of degree µ + b. 3. Main results We first give an expression of the matrix function H (t ). Theorem 3.1. Let G be a graph with adjacency matrix M = an eigenvalue µ of G. Then H (t ) = exp(itM ) = exp(iµt )
A B
B⊤ C
I − B⊤ W (µI − C )−1 B (µI − C )W (µI − C )−1 B
, where C is the adjacency matrix of a star complement for
B⊤ W I − (µI − C )W
,
where W = R−1 (I − exp(−itR)), R = µI − C + (µI − C )−1 BB⊤ . Proof. Notice that exp(−it (µI − M )) = exp(−it µI ) exp(itM ) = exp(−iµt ) exp(itM ), so we get H (t ) = exp(itM ) = exp(iµt ) exp(−it (µI − M )). Since C is the adjacency matrix of a star complement for an eigenvalue µ of G, by Lemma 2.2, we get µI − A = B⊤ (µI − C )−1 B. Hence we have
µI − M = =
µI − A −B
−B⊤ µI − C
I
0 I
(µI − C )−1 B
0 0
−B⊤ R
I
−(µI − C )−1 B
0 , I
J. Zhou, C. Bu / Discrete Applied Mathematics (
)
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3
where R = µI − C + (µI − C )−1 BB⊤ = (µI − C )−1 [(µI − C )2 + BB⊤ ]. Since (µI − C )2 is positive definite, R is nonsingular. Hence we have
µI − M =
µI − M = P
I
0 I
(µI − C )−1 B 0 0
−B⊤
0 0
I
0 , I
−(µI − C )−1 B
R
0 P −1 , R
(1)
where
P = P −1 =
I
0 I
(µI − C )−1 B ⊤ −1 I 0
B R I
−B⊤ R−1
I 0
I I
−(µI − C )−1 B
,
0 . I
Then we can get
exp(−it (µI − M )) =
∞ (−it )n (µI − M )n
=P
=
n!
n=0
I 0
∞ (−it ) P n
n
0 0
0 R
P −1
n!
n=0
0 P −1 . exp(−itR)
By straightforward computation, we have exp(−it (µI − M )) =
I
(µI − C ) B −1
=
0 I
I 0
B⊤ W exp(−itR)
⊤
I
(µI − C )−1 B
I
0 I
−(µI − C ) B −1
B W (µI − C )−1 BB⊤ W + exp(−itR)
I
0 , I
−(µI − C )−1 B
where W = R−1 (I − exp(−itR)), R = µI − C + (µI − C )−1 BB⊤ . Note that
(µI − C )−1 BB⊤ W + exp(−itR) = [R − (µI − C )]W + exp(−itR) = I − (µI − C )W . Hence we have exp(−it (µI − M )) =
(µI − C ) B −1
=
I
B⊤ W I − (µI − C )W
I − B W (µI − C ) B (µI − C )W (µI − C )−1 B ⊤
−1
I
−(µI − C ) B B W . I − (µI − C )W −1
0 I
⊤
From H (t ) = exp(itM ) = exp(iµt ) exp(−it (µI − M )), we can obtain the expression of H (t ).
Theorem 3.2. Let X be a star set for an eigenvalue µ of graph G. Then perfect state transfer does not occur between any two vertices in X . Proof. For any u, v ∈ X , by Lemma 2.2, there exists an eigenvector y of µ such that y⊤ eu ̸= ±y⊤ ev . Lemma 2.1 implies that perfect state transfer does not occur between u and v . For a star set X of a graph G, let X denote the vertex set of the star complement G − X . Theorem 3.3. Let X be a star set for an eigenvalue µ of graph G, and G has adjacency matrix
A B
B⊤ C
, where A is the adjacency
matrix of the subgraph induced by X . If G has perfect state transfer from u to v at time τ and u ∈ X , v ∈ X , then the following statements hold: (1) |(B⊤ W )u,v | = 1, where W = R−1 (I − exp(−iτ R)), R = µI − C + (µI − C )−1 BB⊤ . −1 ⊤ (2) e⊤ v (µI − C ) B = ±eu .
z
Proof. By Theorem 3.1, part (1) holds. By Lemma 2.2, the eigenspace of µ consists of the vectors (µI − C )−1 Bz , where z is −1 ⊤ an arbitrary column vector. Lemma 2.1 implies that e⊤ v (µI − C ) B = ±eu . So part (2) holds.
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J. Zhou, C. Bu / Discrete Applied Mathematics (
)
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Theorem 3.4. Let X be a star set for an eigenvalue µ of graph G, and G has adjacency matrix
A B
B⊤ C
, where A is the adjacency
matrix of the subgraph induced by X . If G has perfect state transfer from u to v at time τ and u, v ∈ X , then the following statements hold: (1) |((µI − C )W )u,v | = 1, where W = R−1 (I − exp(−iτ R)), R = µI − C + (µI − C )−1 BB⊤ . −1 −1 (2) e⊤ B = ±e⊤ u (µI − C ) v (µI − C ) B.
z
Proof. By Theorem 3.1, part (1) holds. By Lemma 2.2, the eigenspace of µ consists of the vectors (µI − C )−1 Bz , where z is −1 −1 an arbitrary column vector. Lemma 2.1 implies that e⊤ B = ±e⊤ u (µI − C ) v (µI − C ) B. So part (2) holds. Let X be a star set for an eigenvalue µ (µ ̸= 0, ±1) of a tree T such that T − X is connected. Then each vertex in X has degree 1 (cf. [12, Proposition 2.1]). Rowlinson proved that the set of all neighbours of vertices in X is also a star set for µ (see [12, Section 2]). Theorem 3.5. Let X be a star set for an eigenvalue µ (µ ̸= 0, ±1) of a tree T such that T − X is connected. Let Y be the set of all neighbours of vertices in X . Then perfect state transfer does not occur between any two vertices in X ∪ Y . Proof. It is known that X and Y are both star sets for eigenvalue µ (see [12, Section 2]). By Theorem 3.2, perfect state transfer does not occur between any two vertices in X or Y . It is also known that each vertex in X has degree 1 (cf. [12, Proposition 2.1]). By µ ̸= 0, ±1, we know that T is a tree with at least 3 vertices. Hence each vertex in Y has degree at least 2. For any u ∈ X , v ∈ Y , u and v are not cospectral vertices. Lemma 2.3 implies that perfect state transfer does not occur from u to v . We use j to denote the all-one vector. Theorem 3.6. Let X be a uniform star set for an eigenvalue µ of graph G, and the star complement G − X is k-regular. For u ∈ X , v ∈ X , perfect state transfer occurs from u to v . Then one of the following hold: (1) µ = k − |Γ (v) ∩ X |. (2) µ = k + |Γ (v) ∩ X |. In this case, µ is a main eigenvalue. Proof. Assume that G has adjacency matrix e⊤ v (µI − −1
A B
B⊤ C
, where A is the adjacency matrix of the subgraph induced by X . By
Theorem 3.3, we have C ) B = ±eu . Since X is a uniform star set, we get Bj = bj, where b = |Γ (v) ∩ X |. ⊤ −1 Then we have e⊤ (µ I − C ) Bj = ±e⊤ j = ±1. Since G − X is k-regular, by Lemma 2.5, we have v u j, bev (µI − C ) ⊤ −1 −1 −1 bev (µI − C ) j = b(µ − k) = ±1. If b(µ − k) = −1, then µ = k − b, part (1) holds. If b(µ − k)−1 = 1, then µ = k + b. If µ = k + b, by Lemma 2.5, B⊤ (µI − C )−1 j = (µ − k)−1 B⊤ j = b−1 B⊤ j ̸= −j. Lemma 2.6 implies that µ = k + b is a main eigenvalue. −1
⊤
Let G be a graph with vertex set V (G), and G has an equitable partition V (G) = V1 ∪ V2 ∪ · · · ∪ Vk . Then each vertex in Vi has dij neighbours in Vj for all i, j ∈ {1, 2, . . . , k}. The matrix (dij )k×k is called the divisor matrix of the equitable partition V (G) = V1 ∪ V2 ∪ · · · ∪ Vk . Theorem 3.7. Let X be a star set for an eigenvalue µ of a connected graph G, and the partition V (G) = X ∪ X is an equitable partition. For u ∈ X , v ∈ X , perfect state transfer occurs from u to v at time τ . Then G is regular and |X | = |X |. Furthermore one of the following hold: (1) µ = 1, G is the path with two vertices. (2) µ = d11 − d21 = d22 − d12 , where exp(2d12 τ i) = −1.
d11 d21
Proof. Assume that G has adjacency matrix
d12 d22
A B
is the divisor matrix of the equitable partition X ∪ X , and d11 = d22 , d12 = d21 , B⊤ C
, where A is the adjacency matrix of the subgraph induced by X . Since
perfect state transfer occurs from u to v at time τ , by Lemma 2.3, u and v are cospectral vertices. So u and v have the same degree. Since X ∪ X is an equitable partition, all vertices in X (or X ) have the same degree. Hence G is regular. From Lemma 2.4 we get |X | = |X |. If µ is the largest eigenvalue of G, since G is connected, we have |X | = |X | = 1. In this case, µ = 1, G is the path with two vertices. Next we only consider the case that µ is not the largest eigenvalue of G. Suppose
d11 d21
d12 d22
is the divisor matrix of
the equitable partition V (G) = X ∪ X , then d11 + d12 = d21 + d22 . Since X is a uniform star set, by Lemma 2.7, we have −1 ⊤ ⊤ −1 d11 = µ + d21 , µ = d11 − d21 = d22 − d12 . By Theorem 3.3, we have e⊤ v (µI − C ) B = ±eu , ev (µI − C ) Bj = −1 ⊤ −1 ⊤ −1 ⊤ ±eu j, d21 ev (µI − C ) j = ±1. By µ = d22 − d12 and Lemma 2.5, we have d21 ev (µI − C ) j = −d21 d12 = ±1, d12 = d21 . From d11 + d12 = d21 + d22 , we get d11 = d22 .
J. Zhou, C. Bu / Discrete Applied Mathematics (
By Lemma 2.4, we have H (τ )
j 0
=α
0 j
)
–
for some scalar α . Since H (τ ) is unitary,
5
j 0
and α
0 j
have the same length.
By |X | = |X |, we get |α| = 1. By Theorem 3.1, there exists a complex number β (|β| = 1) such that
I − B⊤ W (µI − C )−1 B (µI − C )W (µI − C )−1 B
B⊤ W I − (µI − C )W
j 0
=β
0 , j
where W = R−1 (I − exp(−iτ R)), R = µI − C +(µI − C )−1 BB⊤ . So we get j − B⊤ W (µI − C )−1 Bj = 0, (µI − C )W (µI − C )−1 Bj = β j. Hence we have β B⊤ (µI − C )−1 j = j. Recall that G is regular and µ is not the largest eigenvalue of G. It is known that µ is a non-main eigenvalue of G (see [11]). By Lemma 2.6, we get B⊤ (µI − C )−1 j = −j, β = −1. Then
(µI − C )W (µI − C )−1 Bj = −j, W (µI − C )−1 Bj = −(µI − C )−1 j, [I − exp(−iτ R)](µI − C )−1 Bj = −R(µI − C )−1 j = −[I + (µI − C )−1 BB⊤ (µI − C )−1 ]j = −j + (µI − C )−1 Bj − exp(−iτ R)(µI − C )−1 Bj = −j. By µ = d22 − d12 , d12 = d21 and Lemma 2.5, we get 1 (µI − C )−1 Bj = −d21 d− 12 j = −j.
Then we have − exp(−iτ R)(µI − C )−1 Bj = exp(−iτ R)j = −j. Note that Rj = (µI − C )j + (µI − C )−1 BB⊤ j = −d12 j + d12 (µI − C )−1 Bj = −2d12 j Rm j = (−2d12 )m j (m = 0, 1, 2, . . .). By exp(−iτ R)j = −j, we have exp(−iτ R)j =
∞ (−iτ )m Rm j m=0
Hence exp(2d12 τ i) = −1.
m!
=
∞ (−iτ )m (−2d12 )m j m=0
m!
= exp(2d12 τ i)j = −j.
A graph is said to be integral if all its eigenvalues are integers. The research on integral graphs began in 1974 with a paper by Harary and Schwenk (see [9]). Some known graphs with perfect state transfer are integral graphs (see [2,1,14,15]). From Eq. (1) in the proof of Theorem 3.1, we obtain the following sufficient and necessary condition for a graph to be integral. Theorem 3.8. Let G be a graph with adjacency matrix M =
A B
B⊤ C
, where C is the adjacency matrix of a star complement for
an eigenvalue µ1 (µ1 is an integer) of G. Then G is an integral graph if and only if all eigenvalues of R = µ1 I − C +(µ1 I − C )−1 BB⊤ are integers. Acknowledgements The authors would like to thank the reviewers for their valuable suggestions. The research is supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China under grant 11371109. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
M. Bašić, M.D. Petković, Perfect state transfer in integral circulant graphs of non-square-free order, Linear Algebra Appl. 433 (2010) 149–163. M. Bašić, M.D. Petković, D. Stevanović, Perfect state transfer in integral circulant graphs, Appl. Math. Lett. 22 (2009) 1117–1121. A. Bernasconi, C. Godsil, S. Severini, Quantum networks on cubelike graphs, Phys. Rev. A 78 (2008) 5. W.-C. Cheung, C. Godsil, Perfect state transfer in cubelike graphs, Linear Algebra Appl. 435 (2011) 2468–2474. M. Christandl, N. Datta, T. Dorlas, A. Ekert, A. Kay, A. Landahl, Perfect transfer of arbitrary states in quantum spin networks, Phys. Rev. A 71 (2005) 12. D. Cvetković, S. Simić, Graph spectra in computer science, Linear Algebra Appl. 434 (2011) 1545–1562. C. Godsil, Periodic graphs, Electron. J. Combin. 18 (2011) P23. C. Godsil, State transfer on graphs, Discrete Math. 312 (2012) 129–147. F. Harary, A.J. Schwenk, Which graphs have integral spectra? in: R. Bari, F. Harary (Eds.), Graphs and Combinatorics, Proc. Capit. Conf. Graph Theory and Combinatorics, George Washington Univ., June 1973, Springer-Verlag, New York, 1974, pp. 45–51. P. Rowlinson, Star partitions and regularity in graphs, Linear Algebra Appl. 226–228 (1995) 247–265. P. Rowlinson, The main eigenvalues of a graph: a survey, Appl. Anal. Discrete Math. 1 (2007) 445–471. P. Rowlinson, On multiple eigenvalues of trees, Linear Algebra Appl. 432 (2010) 3007–3011. P. Rowlinson, I. Sciriha, Some properties of the Hoffman–Singleton graph, Appl. Anal. Discrete Math. 1 (2007) 438–445. N. Saxena, S. Severini, I. Shparlinski, Parameters of integral circulant graphs and periodic quantum dynamics, Int. J. Quantum Inf. 5 (2007) 417–430. S. Severini, The 3-dimensional cube is the only periodic, connected cubic graph with perfect state transfer, J. Phys.: Conf. Ser. 254 (2010) 012012. D. Stevanović, Applications of graph spectra in quantum physics, in: D. Cvetković, I. Gutman (Eds.), Selected Topics on Applications of Graph Spectra, in: Zb. Rad. (Beogr.), Mathematical Institute SANU, Belgrade, 2011, pp. 85–112. J. Zhou, C. Bu, J. Shen, Some results for the periodicity and perfect state transfer, Electron. J. Combin. 18 (2011) P184.
)
–
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
State transfer and star complements in graphs Jiang Zhou a,b,∗ , Changjiang Bu b a
College of Computer Science and Technology, Harbin Engineering University, Harbin 150001, PR China
b
College of Science, Harbin Engineering University, Harbin 150001, PR China
article
abstract
info
Let G be a graph with adjacency matrix A, and let H (t ) = exp(itA). For an eigenvalue µ of A with multiplicity k, a star set for µ in G is a vertex set X of G such that |X | = k and the induced subgraph G − X does not have µ as an eigenvalue. G is said to have perfect state transfer from the vertex u to the vertex v if there is a time τ such that |H (τ )u,v | = 1. The unitary operator H (t ) has important applications in the transfer of quantum information. In this paper, we give an expression of H (t ). For a star set X of graph G, perfect state transfer does not occur between any two vertices in X . We also give some results for the existence of perfect state transfer in a graph. © 2013 Elsevier B.V. All rights reserved.
Article history: Received 12 June 2012 Received in revised form 24 April 2013 Accepted 19 August 2013 Available online xxxx Keywords: Adjacency matrix Perfect state transfer Star set Star complement Main eigenvalue
1. Introduction Let G be a simple undirected graph with adjacency matrix A. The eigenvalues of A are called the eigenvalues of G. Let H (t ) denote the matrix function H (t ) = exp(itA) =
√
∞ n n n i t A n=0
n!
,
where i = −1. Since A is symmetric, H (t ) is symmetric. Notice that the conjugate transpose of H (t ) is exp(−itA) = H (t )−1 . Hence H (t ) is a unitary matrix. We use Mu,v or (M )u,v to denote the (u, v)-entry of a matrix M. We say that G has perfect state transfer from vertex u to vertex v if there exists a time τ such that |H (τ )u,v | = 1 (see [7,8,16,17]). Some graphs with perfect state transfer are given in [2,1,3–5,7,14,15]. Let G be a graph with an eigenvalue µ of multiplicity k. A star set for µ in G is a vertex set X of G such that |X | = k and the induced subgraph G − X does not have µ as an eigenvalue. In this situation, G − X is called a star complement for µ in G. It is known that star sets exist for any eigenvalue of any graph (see [10,13,12]). The operator H (t ) = exp(itA) has important applications in the transfer of quantum information, it induces a continuoustime quantum walk (see [6,16]). In this paper, we give an expression of H (t ) = exp(itA). For a star set X of graph G, we show that perfect state transfer does not occur between any two vertices in X . We also give some results for the existence of perfect state transfer in a graph. 2. Preliminaries For a vertex u of graph G, we use eu to denote the unit vector whose u-th coordinate is 1 (i.e., eu is the standard basis vector with respect to u). Let A be the adjacency matrix of G. Suppose θ1 , . . . , θm are all distinct eigenvalues of A, and A has
∗
Corresponding author at: College of Computer Science and Technology, Harbin Engineering University, Harbin 150001, PR China. Tel.: +86 82519384. E-mail addresses: [email protected] (J. Zhou), [email protected] (C. Bu).
0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.08.028
2
J. Zhou, C. Bu / Discrete Applied Mathematics (
)
–
spectral decomposition A = r =1 θr Er , where Er is the orthogonal projection on the eigenspace associated with θr . Godsil shows that Er eu = ±Er ev (r = 1, . . . , m) if G perfect state transfer from vertex u to vertex v (cf. [8, Corollary 2.2]). For any eigenvector y of θr , we have y⊤ Er = y⊤ (r = 1, . . . , m). It is not difficult to obtain the following lemma.
m
Lemma 2.1. Let y be an eigenvector of the adjacency matrix of graph G. If G has perfect state transfer from u to v , then y⊤ eu = ±y⊤ ev . Lemma 2.2 ([13,12]). Let X be a set of k vertices in the graph G, and G has adjacency matrix
AX B
B⊤ C
, where AX is the adjacency
matrix of the subgraph induced by X . Then X is a star set for an eigenvalue µ of G if and only if µ is not an eigenvalue of C and
µI − AX = B⊤ (µI − C )−1 B.
z
In this situation, the eigenspace of µ consists of the vectors (µI − C )−1 Bz , where z ∈ Rk is an arbitrary vector. The multiset of eigenvalues of graph G is called the spectrum of G. Vertices u and v in G are called cospectral if G − u and G − v has the same spectrum. Lemma 2.3 ([8]). If graph G has perfect state transfer from u to v , then u and v are cospectral vertices in G. Let G be a graph with vertex set V . The partition V = V1 ∪ V2 ∪ · · · ∪ Vk is an equitable partition if each vertex in the cell Vi has the same number of neighbours in the cell Vj for all i, j ∈ {1, 2, . . . , k}. Lemma 2.4 ([8]). Suppose V1 and V2 are cells of an equitable partition π of graph G, and we have perfect state transfer from u in V1 to v in V2 at time τ . If y is the characteristic vector of the cell V1 , then H (τ )y is the scalar multiple of the characteristic vector of the cell V2 . Furthermore we have |V1 | = |V2 |. Lemma 2.5. Let G be a k-regular graph with adjacency matrix A, and µ is not an eigenvalue of G. Then (µI − A)−1 j = (µ− k)−1 j, where j is the all-one vector. Proof. Since G is k-regular, its largest eigenvalue is k, and the all-one vector j is the eigenvector of k. Then (µI − A)j = (µ − k)j, (µI − A)−1 j = (µ − k)−1 j. An eigenvalue µ of graph G is said to be a main eigenvalue of G if the eigenspace of µ is not orthogonal to the all-one vector (see [11]). It is not difficult to obtain the following lemma from [11, Proposition 2.7]. Lemma 2.6 ([11]). Let G be a graph with adjacency matrix M =
B⊤ C
A B
, where C is the adjacency matrix of a star complement
for an eigenvalue µ of G. Then µ is a non-main eigenvalue if and only if B⊤ (µI − C )−1 j = −j, where j is the all-one vector. We use Γ (v) to denote the set of neighbours of vertex v in a graph. A star set X of graph G is said to be uniform, if there exists a positive integer b such that |Γ (v) ∩ X | = b for every vertex v in the star complement G − X . Lemma 2.7 ([10]). Let X be a uniform star set for an eigenvalue µ of graph G, say |Γ (v) ∩ X | = b for all v in the star complement G − X . If µ is not a main eigenvalue of G, then the graph induced by X is regular of degree µ + b. 3. Main results We first give an expression of the matrix function H (t ). Theorem 3.1. Let G be a graph with adjacency matrix M = an eigenvalue µ of G. Then H (t ) = exp(itM ) = exp(iµt )
A B
B⊤ C
I − B⊤ W (µI − C )−1 B (µI − C )W (µI − C )−1 B
, where C is the adjacency matrix of a star complement for
B⊤ W I − (µI − C )W
,
where W = R−1 (I − exp(−itR)), R = µI − C + (µI − C )−1 BB⊤ . Proof. Notice that exp(−it (µI − M )) = exp(−it µI ) exp(itM ) = exp(−iµt ) exp(itM ), so we get H (t ) = exp(itM ) = exp(iµt ) exp(−it (µI − M )). Since C is the adjacency matrix of a star complement for an eigenvalue µ of G, by Lemma 2.2, we get µI − A = B⊤ (µI − C )−1 B. Hence we have
µI − M = =
µI − A −B
−B⊤ µI − C
I
0 I
(µI − C )−1 B
0 0
−B⊤ R
I
−(µI − C )−1 B
0 , I
J. Zhou, C. Bu / Discrete Applied Mathematics (
)
–
3
where R = µI − C + (µI − C )−1 BB⊤ = (µI − C )−1 [(µI − C )2 + BB⊤ ]. Since (µI − C )2 is positive definite, R is nonsingular. Hence we have
µI − M =
µI − M = P
I
0 I
(µI − C )−1 B 0 0
−B⊤
0 0
I
0 , I
−(µI − C )−1 B
R
0 P −1 , R
(1)
where
P = P −1 =
I
0 I
(µI − C )−1 B ⊤ −1 I 0
B R I
−B⊤ R−1
I 0
I I
−(µI − C )−1 B
,
0 . I
Then we can get
exp(−it (µI − M )) =
∞ (−it )n (µI − M )n
=P
=
n!
n=0
I 0
∞ (−it ) P n
n
0 0
0 R
P −1
n!
n=0
0 P −1 . exp(−itR)
By straightforward computation, we have exp(−it (µI − M )) =
I
(µI − C ) B −1
=
0 I
I 0
B⊤ W exp(−itR)
⊤
I
(µI − C )−1 B
I
0 I
−(µI − C ) B −1
B W (µI − C )−1 BB⊤ W + exp(−itR)
I
0 , I
−(µI − C )−1 B
where W = R−1 (I − exp(−itR)), R = µI − C + (µI − C )−1 BB⊤ . Note that
(µI − C )−1 BB⊤ W + exp(−itR) = [R − (µI − C )]W + exp(−itR) = I − (µI − C )W . Hence we have exp(−it (µI − M )) =
(µI − C ) B −1
=
I
B⊤ W I − (µI − C )W
I − B W (µI − C ) B (µI − C )W (µI − C )−1 B ⊤
−1
I
−(µI − C ) B B W . I − (µI − C )W −1
0 I
⊤
From H (t ) = exp(itM ) = exp(iµt ) exp(−it (µI − M )), we can obtain the expression of H (t ).
Theorem 3.2. Let X be a star set for an eigenvalue µ of graph G. Then perfect state transfer does not occur between any two vertices in X . Proof. For any u, v ∈ X , by Lemma 2.2, there exists an eigenvector y of µ such that y⊤ eu ̸= ±y⊤ ev . Lemma 2.1 implies that perfect state transfer does not occur between u and v . For a star set X of a graph G, let X denote the vertex set of the star complement G − X . Theorem 3.3. Let X be a star set for an eigenvalue µ of graph G, and G has adjacency matrix
A B
B⊤ C
, where A is the adjacency
matrix of the subgraph induced by X . If G has perfect state transfer from u to v at time τ and u ∈ X , v ∈ X , then the following statements hold: (1) |(B⊤ W )u,v | = 1, where W = R−1 (I − exp(−iτ R)), R = µI − C + (µI − C )−1 BB⊤ . −1 ⊤ (2) e⊤ v (µI − C ) B = ±eu .
z
Proof. By Theorem 3.1, part (1) holds. By Lemma 2.2, the eigenspace of µ consists of the vectors (µI − C )−1 Bz , where z is −1 ⊤ an arbitrary column vector. Lemma 2.1 implies that e⊤ v (µI − C ) B = ±eu . So part (2) holds.
4
J. Zhou, C. Bu / Discrete Applied Mathematics (
)
–
Theorem 3.4. Let X be a star set for an eigenvalue µ of graph G, and G has adjacency matrix
A B
B⊤ C
, where A is the adjacency
matrix of the subgraph induced by X . If G has perfect state transfer from u to v at time τ and u, v ∈ X , then the following statements hold: (1) |((µI − C )W )u,v | = 1, where W = R−1 (I − exp(−iτ R)), R = µI − C + (µI − C )−1 BB⊤ . −1 −1 (2) e⊤ B = ±e⊤ u (µI − C ) v (µI − C ) B.
z
Proof. By Theorem 3.1, part (1) holds. By Lemma 2.2, the eigenspace of µ consists of the vectors (µI − C )−1 Bz , where z is −1 −1 an arbitrary column vector. Lemma 2.1 implies that e⊤ B = ±e⊤ u (µI − C ) v (µI − C ) B. So part (2) holds. Let X be a star set for an eigenvalue µ (µ ̸= 0, ±1) of a tree T such that T − X is connected. Then each vertex in X has degree 1 (cf. [12, Proposition 2.1]). Rowlinson proved that the set of all neighbours of vertices in X is also a star set for µ (see [12, Section 2]). Theorem 3.5. Let X be a star set for an eigenvalue µ (µ ̸= 0, ±1) of a tree T such that T − X is connected. Let Y be the set of all neighbours of vertices in X . Then perfect state transfer does not occur between any two vertices in X ∪ Y . Proof. It is known that X and Y are both star sets for eigenvalue µ (see [12, Section 2]). By Theorem 3.2, perfect state transfer does not occur between any two vertices in X or Y . It is also known that each vertex in X has degree 1 (cf. [12, Proposition 2.1]). By µ ̸= 0, ±1, we know that T is a tree with at least 3 vertices. Hence each vertex in Y has degree at least 2. For any u ∈ X , v ∈ Y , u and v are not cospectral vertices. Lemma 2.3 implies that perfect state transfer does not occur from u to v . We use j to denote the all-one vector. Theorem 3.6. Let X be a uniform star set for an eigenvalue µ of graph G, and the star complement G − X is k-regular. For u ∈ X , v ∈ X , perfect state transfer occurs from u to v . Then one of the following hold: (1) µ = k − |Γ (v) ∩ X |. (2) µ = k + |Γ (v) ∩ X |. In this case, µ is a main eigenvalue. Proof. Assume that G has adjacency matrix e⊤ v (µI − −1
A B
B⊤ C
, where A is the adjacency matrix of the subgraph induced by X . By
Theorem 3.3, we have C ) B = ±eu . Since X is a uniform star set, we get Bj = bj, where b = |Γ (v) ∩ X |. ⊤ −1 Then we have e⊤ (µ I − C ) Bj = ±e⊤ j = ±1. Since G − X is k-regular, by Lemma 2.5, we have v u j, bev (µI − C ) ⊤ −1 −1 −1 bev (µI − C ) j = b(µ − k) = ±1. If b(µ − k) = −1, then µ = k − b, part (1) holds. If b(µ − k)−1 = 1, then µ = k + b. If µ = k + b, by Lemma 2.5, B⊤ (µI − C )−1 j = (µ − k)−1 B⊤ j = b−1 B⊤ j ̸= −j. Lemma 2.6 implies that µ = k + b is a main eigenvalue. −1
⊤
Let G be a graph with vertex set V (G), and G has an equitable partition V (G) = V1 ∪ V2 ∪ · · · ∪ Vk . Then each vertex in Vi has dij neighbours in Vj for all i, j ∈ {1, 2, . . . , k}. The matrix (dij )k×k is called the divisor matrix of the equitable partition V (G) = V1 ∪ V2 ∪ · · · ∪ Vk . Theorem 3.7. Let X be a star set for an eigenvalue µ of a connected graph G, and the partition V (G) = X ∪ X is an equitable partition. For u ∈ X , v ∈ X , perfect state transfer occurs from u to v at time τ . Then G is regular and |X | = |X |. Furthermore one of the following hold: (1) µ = 1, G is the path with two vertices. (2) µ = d11 − d21 = d22 − d12 , where exp(2d12 τ i) = −1.
d11 d21
Proof. Assume that G has adjacency matrix
d12 d22
A B
is the divisor matrix of the equitable partition X ∪ X , and d11 = d22 , d12 = d21 , B⊤ C
, where A is the adjacency matrix of the subgraph induced by X . Since
perfect state transfer occurs from u to v at time τ , by Lemma 2.3, u and v are cospectral vertices. So u and v have the same degree. Since X ∪ X is an equitable partition, all vertices in X (or X ) have the same degree. Hence G is regular. From Lemma 2.4 we get |X | = |X |. If µ is the largest eigenvalue of G, since G is connected, we have |X | = |X | = 1. In this case, µ = 1, G is the path with two vertices. Next we only consider the case that µ is not the largest eigenvalue of G. Suppose
d11 d21
d12 d22
is the divisor matrix of
the equitable partition V (G) = X ∪ X , then d11 + d12 = d21 + d22 . Since X is a uniform star set, by Lemma 2.7, we have −1 ⊤ ⊤ −1 d11 = µ + d21 , µ = d11 − d21 = d22 − d12 . By Theorem 3.3, we have e⊤ v (µI − C ) B = ±eu , ev (µI − C ) Bj = −1 ⊤ −1 ⊤ −1 ⊤ ±eu j, d21 ev (µI − C ) j = ±1. By µ = d22 − d12 and Lemma 2.5, we have d21 ev (µI − C ) j = −d21 d12 = ±1, d12 = d21 . From d11 + d12 = d21 + d22 , we get d11 = d22 .
J. Zhou, C. Bu / Discrete Applied Mathematics (
By Lemma 2.4, we have H (τ )
j 0
=α
0 j
)
–
for some scalar α . Since H (τ ) is unitary,
5
j 0
and α
0 j
have the same length.
By |X | = |X |, we get |α| = 1. By Theorem 3.1, there exists a complex number β (|β| = 1) such that
I − B⊤ W (µI − C )−1 B (µI − C )W (µI − C )−1 B
B⊤ W I − (µI − C )W
j 0
=β
0 , j
where W = R−1 (I − exp(−iτ R)), R = µI − C +(µI − C )−1 BB⊤ . So we get j − B⊤ W (µI − C )−1 Bj = 0, (µI − C )W (µI − C )−1 Bj = β j. Hence we have β B⊤ (µI − C )−1 j = j. Recall that G is regular and µ is not the largest eigenvalue of G. It is known that µ is a non-main eigenvalue of G (see [11]). By Lemma 2.6, we get B⊤ (µI − C )−1 j = −j, β = −1. Then
(µI − C )W (µI − C )−1 Bj = −j, W (µI − C )−1 Bj = −(µI − C )−1 j, [I − exp(−iτ R)](µI − C )−1 Bj = −R(µI − C )−1 j = −[I + (µI − C )−1 BB⊤ (µI − C )−1 ]j = −j + (µI − C )−1 Bj − exp(−iτ R)(µI − C )−1 Bj = −j. By µ = d22 − d12 , d12 = d21 and Lemma 2.5, we get 1 (µI − C )−1 Bj = −d21 d− 12 j = −j.
Then we have − exp(−iτ R)(µI − C )−1 Bj = exp(−iτ R)j = −j. Note that Rj = (µI − C )j + (µI − C )−1 BB⊤ j = −d12 j + d12 (µI − C )−1 Bj = −2d12 j Rm j = (−2d12 )m j (m = 0, 1, 2, . . .). By exp(−iτ R)j = −j, we have exp(−iτ R)j =
∞ (−iτ )m Rm j m=0
Hence exp(2d12 τ i) = −1.
m!
=
∞ (−iτ )m (−2d12 )m j m=0
m!
= exp(2d12 τ i)j = −j.
A graph is said to be integral if all its eigenvalues are integers. The research on integral graphs began in 1974 with a paper by Harary and Schwenk (see [9]). Some known graphs with perfect state transfer are integral graphs (see [2,1,14,15]). From Eq. (1) in the proof of Theorem 3.1, we obtain the following sufficient and necessary condition for a graph to be integral. Theorem 3.8. Let G be a graph with adjacency matrix M =
A B
B⊤ C
, where C is the adjacency matrix of a star complement for
an eigenvalue µ1 (µ1 is an integer) of G. Then G is an integral graph if and only if all eigenvalues of R = µ1 I − C +(µ1 I − C )−1 BB⊤ are integers. Acknowledgements The authors would like to thank the reviewers for their valuable suggestions. The research is supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China under grant 11371109. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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