Laplacian Fractional Revival on Graphs

We develop the theory of fractional revival in the quantum walk on a graph using its Laplacian matrix as the Hamiltonian. We first give a spectral characterization of Laplacian fractional revival, which leads to a polynomial time algorithm to check this phenomenon and find the earliest time when it occurs. We then apply the characterization theorem to special families of graphs. In particular, we show that no tree admits Laplacian fractional revival except for the paths on two and three vertices, and the only graphs on a prime number of vertices that admit Laplacian fractional revival are double cones. Finally, we construct, through Cartesian products and joins, several infinite families of graphs that admit Laplacian fractional revival; some of these graphs exhibit polygamous fractional revival.


Introduction
A quantum walk on a graph X is determined by the transition operator where H is some Hermitian matrix indexed by the vertices. As noted by Bose et al. [4], the two common quantum walks, where H is the adjacency or Laplacian matrix of X, are related to spin networks in the XY and XYZ interaction models, respectively.
We say X admits fractional revival at time τ from vertex a to vertex b if U (τ ) = αe a + βe b for some complex numbers α and β. Fractional revival, although rare, is a useful phenomenon in transmitting quantum information [3]. One of its special cases, perfect state transfer, has been extensively studied from both physical and combinatorial viewpoints. In particular, one can decide adjacency perfect state transfer in polynomial time [8], using the spectral information of A. There are also characterizations of Laplacian perfect state transfer or adjacency fractional revival on graphs that are combinatorially interesting. For example, Coutinho and Liu [9] ruled out trees on more than two vertices for Laplacian perfect state transfer, and Chan et al. [5] characterized several families of distance regular graphs with adjacency fractional revival.
Laplacian fractional revival, on the other hand, has not been studied in great detail. There is a characterization of threshold graphs that admit Laplacian fractional revival, due to Kirkland and Zhang [13], but a general framework, which allows us to investigate this phenomenon systematically, is missing from the literature.
In this paper, we prove necessary and sufficient conditions (Theorem 7.6) for proper Laplacian fractional revival to occur on a general graph. As a consequence, we arrive at a polynomial time algorithm (Theorem 9.1) that decides Laplacian fractional revival, and if so, finds the earliest time when it occurs.
We then proceed to determine Laplacian fractional revival on special classes of graphs, including trees (Section 10) and join graphs (Section 11). In particular, we prove no tree on more than three vertices admits proper Laplacian fractional revival (Theorem 10.3), which implies the result in [9]. We also show that every graph on a prime number of vertices that admits proper Laplacian fractional revival is a double cone (Theorem 11.9).
Finally, we use Cartesian products and joins to construct several infinite families of graphs (Theorem 5.3, Theorem 11.6 and Theorem 12.2) with proper fractional revival. On the last family, this phenomenon behaves somewhat surprisingly, as it occurs from a to b and from a to c with b = c. This is in stark contrast to perfect state transfer, for which a must be paired with a unique vertex. As these graphs are regular, they are also examples of unweighted graphs that admit polygamous adjacency fractional revival, which answers an open question in [6].
Moreover, the multiplicity of n is one less than the number of components of the complement X.
Given an orientation of X, the signed incidence matrix, denoted B, is the matrix whose rows are indexed by the vertices and columns by the edges with if v is the head of the edge e −1, if v is the tail of edge e 0, otherwise The following lemma shows a connection between the Laplacian matrix and the signed incidence matrices of a graph.   Tree Theorem). Let X be a graph on n vertices with Laplacian matrix L. Let q be the number of spanning trees of X. If u is a vertex of X, then det(L[u|u]) = q. Moreover, nq equals the product of non-zero eigenvalues of L.

Laplacian fractional revival
Given a Hermitian matrix H, a quantum walk with respect to H is determined by the following matrix, called the transition matrix: Note that U (t) is a unitary matrix. If, in addition, H is real symmetric, then U (t) is also symmetric.
There are two common choices for H associated with a graph: the adjacency matrix A and the Laplacian matrix L. In this paper we will consider the latter. However, for a d-regular graph, we have L = dI − A, and so exp(itL) = e idt exp(−itA). Therefore, some of our results apply to the adjacency model as well if the graph is regular.
We say a graph admits Laplacian fractional revival, or LaFR, from vertex a to vertex b at time τ if the transition matrix with respect to L satisfies U (τ )e a = αe a + βe b for some complex numbers α and β. As U (t) is unitary, it follows that |α| 2 + |β| 2 = 1. Equivalently, LaFR occurs if U (τ ) has the following block diagonal form: There are some special cases of LaFR. If β = 0, then the graph is said to be Laplacian periodic at vertex a at time τ ; in this case, vertex b does not play a role. We will refer to the case where β = 0 as proper LaFR. If, on the other hand, α = 0, then the graph is said to admit Laplacian perfect state transfer, or LaPST, from vertex a to vertex b.
We can compute U (t) using the eigenvalues and eigenprojections of L. Let the spectral decomposition of L be where µ r is an eigenvalue of L, and F r is the orthogonal projection onto the eigenspace of µ r . Then we have This formula will be frequently used in the rest of the paper.
We list some examples that admit LaFR.
Example 3.1. K 2 admits three types of LaFR: (i) At time t ∈ πZ, it is Laplacian periodic at both vertices.
(ii) At time t ∈ π 2 Z, it has LaPST from one vertex to the other. (iii) At time t / ∈ π 2 Z, it has proper LaFR from one vertex to the other. Example 3.2. On P 3 , there is proper LaFR at time 2π/3 between the end vertices, and periodicity at time 2π at all vertices.
The next example takes a look at four graphs that admit proper LaFR in a very interesting way. Example 3.3. In Figure 3, the vertices that are not black in each graph are the ones between which proper LaFR occurs. For example, in the first graph, C 6 , proper LaFR occurs between antipodal vertices. Comparing the first to the second graph, we can see that adding an edge between two vertices that do not have LaFR destroys their LaFR with their matching vertices. Going from the second to the fourth graph still preserves the LaFR. Now, what is particularly interesting is the comparison between the second and the third graph: Adding an edge between the two vertices admitting proper LaFR still preserves the proper LaFR. All of these are just observations on these specific graphs and do not apply to all graphs.

Cartesian products
In this section, we discuss a graph operation that preserves LaFR.
Let X and Y be two graphs. The Cartesian product of X and Y , denoted X Y , is the graph with vertex set V (X) × V (Y ), such that two vertices (x 1 , y 1 ) and (x 2 , y 2 ) are adjacent if either x 1 is adjacent to x 2 in X and y 1 = y 2 , or x 1 = x 2 and y 1 is adjacent to y 2 . Using the Kronecker product of matrices, we can express the Laplacian matrix of X Y in terms of the Laplacian matrices of X and Y : for details about this identity, see Godsil and Coutinho [7]. As L(X) ⊗ I commutes with I ⊗ L(Y ), it follows that We characterize proper LaFR on Cartesian products in terms of their base graphs. Proof. First, suppose X Y has proper LaFR at time τ . Then Let a ij and b ij be the ij-entries of U X (τ ) and U Y (τ ), respectively. We have and for j ≥ 3, As LaFR on X Y is proper, β = 0, and so a 11 = 0. From (2) it follows that a 11 = 0, b 12 = 0 and b 21 = 0. Further, from (3) we can conclude that So, Y indeed has proper LaFR at time τ . Now, since b 12 = 0 and b 21 = 0, by similar reasoning we have a j1 = a 1j = 0 for all j = 1.Therefore, X must have a periodic vertex at τ . Conversely, suppose at time τ , X has a periodic vertex and Y has proper LaFR. Then As LaFR on Y is proper, β = 0, and so LaFR on X Y is proper.

Complements and joins
Under certain conditions, LaFR is also preserved by taking complements or joins. The following result generalizes Theorem 2 of [1].
Theorem 5.1. Let X be a graph on n vertices. If X has LaPST or LaFR between two vertices u and v at time τ , with nτ ∈ 2πZ, then the complement X has LaPST or LaFR, respectively, between these two vertices at τ .
Proof. Let L be the Laplacian matrix of X, and L the Laplacian matrix of X. We have L = nI − J − L.
Given two graphs X and Y , the join of X and Y , denoted X + Y , is the graph obtained from the disjoint union X ∪ Y by joining all vertices of X to all vertices of Y . Equivalently, Corollary 5.2. Let X be a graph whose complement admits LaPST or LaFR between vertex a and vertex b at time τ . For any graph Y , the join X +Y has LaPST or LaFR, respectively, between these two vertices at time τ provided that τ (|V (X)| + |V (Y )|) ∈ 2πZ.
Proof. This follows from the above identity and Theorem 5.1.
As a special case of joins, the graph K 2 + Y is called the double cone over Y . We will refer to the two vertices from K 2 in a double cone as the conical vertices. Our next result shows that double cones provide an infinite family of graphs on which proper LaFR occurs. Proof. Let K 2 + Y be a double cone on n vertices, and let τ = 2π/n. As n ≥ 3, we have τ / ∈ πZ, and so by Example 3.1, K 2 has proper LaFR at time τ . Now apply Corollary 5.2 with X = K 2 .
Since LaPST occurs on K 2 at odd multiples of π/2, the double cone K 2 + Y admits LaPST at time 2π/n if and only if n = 4.
At the end of this section, we mention another infinite family of graphs, found by Kirkland and Zhang [13], with LaFR.

Laplacian strong cospectrality
In this section, we study a spectral property called Laplacian strong cospectrality. This is a key property required by proper LaFR.
Let X be a graph. Let L be its Laplacian matrix with spectral decomposition L = r µ r F r .
We say two vertices a and b are Laplacian strongly cospectral if for each r, This extends the notion of adjacency strong cospectrality due to Godsil and Smith [12]. It is proved that if a graph admits perfect state transfer between two vertices a and b, then they must be strongly cospectral. In the next section, we will show that Laplacian strong cospectrality is also a necessary condition for LaFR. Given two strongly cospectral vertices a and b with respect to L, there is a natural partition of the eigenvalues of L: This leads to the following observation. Proof. Let the spectral decomposition of L be The result follows as the diagonal entries of L are degrees of the vertices.
Our next lemma gives the structure of the spectral idempotents summed over each class.
Suppose vertices a and b are strongly cospectral with respect to L. Then As the spectral idempotents F r sum to the identity, we have which corresponds to the first columns of forms of their second column and the rest of their first two rows follows from strong cospectrality and that the matrices are symmetric.
If a and b are Laplacian strongly cospectral, then the eigenvalue 0 lies in Φ ab . By the two identities in Lemma 6.2, Φ + ab contains at least one non-zero eigenvalue, and Φ − ab cannot be empty. Hence we have the following lower bounds. Corollary 6.3. Let X be a graph with at least three vertices. If a, b are Laplacian strongly cospectral vertices in X, then |Φ + ab | ≥ 2 and |Φ − ab | ≥ 1. We can say more about this partition. Proof. Let µ be an eigenvalue which is not an integer and v any of its eigenvectors. Let µ ′ be an algebraic conjugate of µ. There is a field automorphism Ψ of Q(µ) that maps µ to µ ′ . It also maps v to a vector v ′ , which is an eigenvector for the eigenvalue µ ′ . As a and b are strongly cospectral, it follows that v a = ±v b .
We have seen that Laplacian strongly cospectral vertices have the same degree. In the following theorem, we give an upper bound and lower bound for this degree. Theorem 6.5. Let X be a connected graph on n vertices. Let a and b be two strongly cospectral vertices with respect to the Laplacian matrix. Let θ ± denote the largest element in Φ ± ab , and λ ± the smallest non-zero element in Φ ± ab . Then the following hold.
(i) If a and b are not adjacent, then As a and b are Laplacian strongly cospectral, for each eigenvalue µ r ∈ Φ + ab , there is a non-zero constant c r such that and for each eigenvalue µ r ∈ Φ − ab , there is a non-zero constant c r such that On the other hand, if d = deg(a) then where σ = 1 if a and b are adjacent, and σ = 0 otherwise. We abuse notation and denote r : µ r ∈ Φ ± ab by r ∈ Φ ± ab for convenience. Hence by we obtain two equalities: Now, recall that 0 is an eigenvalue in Φ + ab with eigenvector 1. Moreover, by Lemma 6.2, Therefore Equation 5 tells us that Likewise, Equation (6) tells us that The upper bound for d follows from a similar argument.
Our next result links the integer eigenvalues in Φ − ab to the number of spanning trees of the graph. Lemma 6.6. Let X be a graph on n vertices with Laplacian strongly cospectral vertices a and b. Let µ ∈ Φ − ab be an integer. Then any odd prime p that divides µ must divide the number of spanning trees of X.
Proof. Let q be the number of spanning trees of X. By the Matrix-Tree Theorem, any (n − 1) × (n − 1) minor of the Laplacian matrix L equals q. Therefore, for each odd prime p not dividing q, the Laplacian matrix has rank n − 1 over Z p , with kernel being the span of 1.
Now let a and b be two strongly cospectral vertices with respect to L. Suppose, for a contradiction, that some integer eigenvalue µ ∈ Φ − ab has an odd prime factor p that does not divide q. Let x be an eigenvector of µ, and assume without loss of generality that the gcd of its entries is 1. Then since there is an integer k such that As µ ∈ Φ − ab , the projection of e a and e b onto any subspace of the µ-eigenspace must be opposites. Therefore x a = −x b . Hence k ≡ −k (mod p). But as p is an odd prime, k is divisible by p. Hence p divides all entries of x. This contradicts the assumption that all entries of x have gcd 1.
Finally, we show a connection between the size of Φ − ab and the distance between a and b.
Theorem 6.7. Let a and b be Laplacian strongly cospectral vertices in X. If |Φ − ab | = k, then any vertex at distance k from a is at distance at most k from b. In particular, the distance between a and b is no more than 2k.
Proof. Define as in Lemma 6.2 Then we have W v − = 0. Hence W e a = W e b . Now, notice that W is the product of k matrices in the form (D i − A), where A is the adjacency matrix and each D i is some diagonal matrix. By the expanded form of W , the ij-entry is non-zero only if d(i, j) ≤ k, and is ±1 if d(i, j) = k. Since the a-th column and the b-th column of W are identical, we conclude that any vertex at distance k from a is at distance at most k from b.
As a consequence, we have proved the following result due to Coutinho and Liu [9]. Two vertices are called twins if they have the same neighbors.

Proper Laplacian fractional revival
We now characterize proper LaFR on connected graphs. The following lemma shows that α = γ in Equation (1), provided β = 0. This is an important observation as it leads to Laplacian strong cospectrality.
Lemma 7.1. If X admits proper LaFR between a and b at time τ , then U (τ ) aa = U (τ ) bb . Moreover, each spectral idempotent F r of L satisfies (F r ) aa = (F r ) bb .
Proof. We already know that Let M be the submatrix of U (τ ) indexed by a and b. Since β = 0, M has two distinct eigenvalues. Moreover, as 1 is an eigenvector of L, the eigenvectors of M must be 1 1 , 1 −1 .
For the second statement, let the spectral decomposition of L be Then r e itµr F r = M 0 0 N .
Multiply both sides by F r and we get As before, let F r denote the submatrix of F r indexed by a and b. We have from which it follows that the column space of F r is a subspace of an eigenspace of M . Therefore, (F r ) aa = (F r ) bb .
We are now ready to show that proper LaFR can only happen between Laplacian strongly cospectral vertices. For each r, multiplying both sides by F r yields e itµr F r e a = αF r e a + βF r e b .
As β is non-zero, F r e b is a scalar multiple of F r e a . Finally, Lemma 7.1 tells us that (F r ) aa = (F r ) bb . Hence it must be that F r e a = ±F r e b .
Our next result shows that each eigenvalue in Φ + ab or Φ − ab respects the class it belongs to. It becomes useful when we try to determine the time when proper LaFR occurs. respectively. Now for each r, If F r denotes the submatrix of F r indexed by a and b, then that is, the column space of F r is a subspace of an eigenspace of M . On the other hand, we know from the proof of Theorem 6.5 that each F r is rank-one; more specifically, if µ r ∈ Φ + ab , Thus, if µ r and µ s both lie in Φ + ab or both lie in Φ − ab , then F r and F s are scalar multiples of each other, and so e iτ µr = e iτ µs .
As a consequence, we get a ratio condition on the eigenvalues when proper LaFR occurs.
Proof. For µ i , µ j in the same Φ + ab and Φ − ab class, e iτ µ i = e iτ µ j . Hence The result follows by taking the ratio of τ (µ i − µ j ) and τ (µ r − µ s ).
We cite a powerful result due to Godsil [10, Theorem 6.1]; it bounds the algebraic degrees of the elements in Φ + ab and Φ − ab .
[10] Let Φ be a set of real algebraic integers which is closed under taking algebraic conjugates. Suppose for all µ i , µ j , µ r , µ s ∈ Φ with µ r = µ s , we have Then the elements of Φ are either integers or quadratic integers, and, moreover, if |Φ| = n, then there are integers a, ∆ (square-free), and {b r } n r=1 such that µ ∈ Φ implies that, for some r, With all the theory we developed so far, we are now ready to give a characterization of proper LaFR.
proper LaFR occurs between a and b if and only if the following conditions hold: (i) a and b are strongly cospectral vertices.
ab that is not divisible by g. Moreover, if proper LaFR occurs between a and b at time τ , then τ is an integer multiple of 2π/g.

Proof.
We first prove the if direction. Let τ = 2π/g. By the definition of g, the function µ r → e iτ µr is constant within Φ + ab and within Φ − ab . Let µ s be an element in Φ − ab that is not divisible by g. Then e iτ µs = 1. Now, noticing 0 ∈ Φ + ab and using Lemma 6.2, For the only if direction, suppose proper LaFR occurs between a and b. Condition (i) follows from Theorem 7.2. Using Corollary 7.4, Theorem 6.4 and Theorem 7.5, we have integers a + , a − , and square free integers ∆ + , ∆ − , and {b r } d r=0 , such that, for all r = 0, .., d, , and Then τ is an integer multiple of 2π/g + √ ∆ + and also an integer multiple of 2π/g − √ ∆ − . Therefore ∆ + = ∆ − . Let ∆ = ∆ + . We now prove Condition (ii). By Corollary 6.3, Φ + ab contains at least two elements. Since 0 ∈ Φ + a,b , we must have a + = 0. As Φ + ab is closed under taking algebraic conjugates, if ∆ = 1, then for any . However, this contradicts the fact L is positive semidefinite, as all eigenvalues of L should be non-negative. Therefore ∆ = 1, and so Φ + a,b ∪ Φ − a,b ⊆ Z ≥0 . Finally, by Lemma 7.3, LaFR can only occur at at times 2π g Z, where g is defined as in this theorem. If µ r → e iτ µr is constant on the entire set Φ + ab ∪ Φ − ab , then the X is periodic with respect to L. Hence, there must be some eigenvalue in Φ − a,b that is not divisible by g. A direct consequence of this result is that the vertices involved in proper LaFR must have degree at least two, unless the graph has fewer than five vertices. Proof. Since a and b are Laplacian strongly cospectral, by Theorem 6.5, we have where λ + is the smallest element in Φ + ab , and σ = 1 if a is adjacent to b and σ = 0 otherwise. As the LaFR is proper, g = 1, and so λ ≥ 2. The result then follows when n ≥ 5.

Laplacian periodicity
In this section, we characterize Laplacian periodicity at a vertex a of a graph. This can be viewed as non-proper LaFR; however, it is also a necessary condition for proper LaFR to occur from a to another vertex.
As before, let X be a graph with Laplacian matrix L, and suppose the spectral decomposition is The eigenvalue support of a vertex a, denoted Φ a , is the set Φ a = {µ r : F r e a = 0}.
Since F r is positive semidefinite, equivalently, Φ a = {µ r : (F r ) aa = 0.} We will let Φ 0 a denote the complement of Φ a . Define two polynomials and We can compute Φ a using ψ and ψ a . By Cramer's Rule and the spectral decomposition of L, This leads to the following observation.
Lemma 8.1. The elements in Φ a are precisely the poles of ψ a (t)/ψ(t).
We also note an interesting connection between Φ a and the spanning trees of X. Suppose X has n vertices and deg(a) = d. Let µ r be an eigenvalue of L with eigenvector x. If x a = 0, then we may put the Laplacian matrix in the form where the last row is indexed by a. Then the restriction of x to X \a, denoted x, is an eigenvector of L[a|a] with eigenvalue µ r . Hence, the restriction of vectors in the µ r -eigenspace to X\a, if non-zero, is an eigenvector for L[a|a] with eigenvalue µ r .
On the other hand, if we let W denote the subspace R n−1 × {0}, then µ ∈ Φ 0 a if and only if the µ r -eigenspace is a subspace of W , and so Therefore, if µ r ∈ Φ 0 a , then µ r is an eigenvalue of L[a|a] of same or higher multiplicity, and if µ r ∈ Φ a , then µ r is an eigenvalue of L[a|a] with multiplicity at least one less. By the Matrix-Tree theorem, we arrive at the following.
Theorem 8.2. Let X be a graph and let a be a vertex. The product of elements in Φ 0 a divides the number of spanning trees of X. Theorem 7.6 gives a characterization of proper LaFR between a and b using Φ + ab and Φ − ab . We now use Φ a to characterize Laplacian periodicity. The proof is very similar, so we omit it here.
Clearly, if a is strongly cospectral to b, then Φ a = Φ + ab ∪ Φ − ab , and Φ 0 a = Φ 0 ab . Thus we have the following corollary. In [10], Godsil showed that for any integer k, there are only finitely many connected graphs with maximum valency that admit adjacency perfect state transfer. We extend his result to Laplacian periodicity.
Given a graph X and a vertex a, the eccentricity of a, denoted ecc(a), is the maximum distance from a to any other vertex in X. We show that the size of Φ a determines an upper bound for ecc(a). Then M is a weighted adjacency matrix of the graph with non-negative entries. Thus, for any i < j, the support of M i e a is a proper subset of the support of M j e a , and so the vectors e a , M e a , M 2 e a , · · · , M ℓ e a are linearly independent. On the other hand, M is a linear combination of the spectral idempotents F r of L, so span{e a , M e a , M 2 e a , · · · , M ℓ e a } ⊆ span{F r e a : r ∈ Φ a }.
Using the second upper bound in Theorem 2.2, we are able to prove the following theorem. It explains why Laplacian periodicity, and hence LaFR, is a rare phenomenon.
Theorem 8.6. Given an integer k, there are only finitely many connected graphs with maximum valency at most k that are Laplacian periodic at a vertex.
Proof. Let X be a graph with maximum valency k. Suppose X is Laplacian periodic at a vertex a. Then Φ a contains only integers. By Theorem 2.2, the eigenvalues of L are no greater than 2k. Hence, 2k + 1 ≥ |Φ a | ≥ ecc(a) + 1, from which we have ecc(a) ≤ 2k, and only finitely many graphs satisfy this constraint.

Laplacian fractional revival is polynomial time
Using the theory we developed so far, we show that LaFR can be decided in polynomial time.
Theorem 9.1. Deciding whether a graph has proper LaFR, and the earliest time when it occurs, can be done in polynomial time.
Proof. We prove this by showing that all three conditions in Theorem 7.6 can be checked in polynomial time.
By lemma 2.4 in [8], Laplacian strong cospectrality, which is Condition (i), can be checked in polynomial time. Now we adapt lemma 2.5 in [8] to proper LaFR. By Lemma 8.1, the elements in Φ a are precisely the poles of ψ a (t)/ψ(t), which are all simple. Equivalently, Φ a consists of simple roots of To find integer eigenvalues in Φ a , recall that all eigenvalues of L lie in [0, n], so we simply check whether 0, 1, · · · , n are roots of f (x), which can be done in polynomial time. Moreover, if f (x) has degree k, then the coefficient of (−x) k−1 is the sum of the roots of f (x). By comparing the sum of the roots we found with the coefficient of (−x) k−1 , we can decide whether all eigenvalues in Φ a are integers.
We then use Gaussian elimination to calculate the corresponding eigenvectors in polynomial time. This means we can decide in polynomial time how to partition Φ a into Φ + ab and Φ − ab . Therefore, Condition (ii) can be checked in polynomial time.
Finally, let g = gcd µ r − µ s : µ r , µ s ∈ Φ + ab or µ r , µ s ∈ Φ − ab . Since there are at most n eigenvalues in Φ + ab ∪ Φ − ab , the set over which we take the gcd has size at most n 2 = O(n 2 ). Moreover, as all elements in the set live in [0, n], we can compute g in polynomial time. It remains to check no element in Φ − ab , which has size less than n, is divisible by g. Hence, Condition (iii) can be checked in polynomial time.

No proper Laplacian fractional revival on trees
In this section, we show that proper LaFR does not occur on trees except for K 2 and P 3 . An earlier result about LaPST on trees, due to Coutinho and Liu [9], can be viewed as a consequence.
We first prove a technical lemma on the signed incidence matrices of trees.
Lemma 10.1. Let T be a tree on n vertices. Fix an orientation of T , and let B be the signed incidence matrix. If µ ≥ 2 is an integer, then any solution to By ≡ 0 (mod µ) must satisfy y ≡ 0 (mod µ).
Proof. If T = K 2 , then and from By ≡ 0 (mod µ) it follows that both entries of y are divisible by µ. Now let T be a general tree on n ≥ 3 vertices. Let v be a leaf of the tree, and e the edge incidence to v. Then By ≡ 0 (mod µ) implies that y e is divisible by µ. Thus, if y ′ denotes the restriction of y to T \v, then Note that B[v|e] is the signed incidence matrix of some orientation of T \v. By induction, we see that y ≡ 0 (mod µ).
As L = BB T , the above lemma imposes number theoretic conditions on the eigenvectors of L associated with integer eigenvalues. On the other hand, for every edge e = (u, v) of X, we have y e = x u − x v . Thus x u − x v is divisible by µ. As T is connected, the difference of any two entries of x are divisible by µ.
We now prove the main non-existence result about proper LaFR on trees.
Theorem 10.3. The only trees the admit proper LaFR are K 2 and P 3 .
Proof. Let T be a tree with proper LaFR between a and b. If |Φ − ab | = 1, then a and b are twin vertices by Corollary 6.8. As T is a tree, there can only be one common neighbor of a and b, and so they are leaves. Thus by Corollary 7.7, T has at most four vertices. It is easy to check that only K 2 and P 3 admit LaFR. Now suppose |Φ − ab | ≥ 2. Let µ ∈ Φ − ab be an eigenvalue of L with eigenvector x. Since µ is an integer, we may assume without loss of generality that x has integer entries, and that the gcd of these entries is 1. By Corollary 10.2, However, as µ ∈ Φ − ab , we also have x u = −x v . Hence µ divides 2x u . Moreover, if gcd(µ, x u ) were not 1, then it would appear as a common factor of all entries of x, which contradicts our assumption. Therefore µ divides 2. Since |Φ − ab | ≥ 2, we must have Φ − ab = {1, 2}. By Theorem 7.6, LaFR occurs at time 2π/g with g = 1, and this cannot be proper.

Laplacian fractional revival on joins
We derive more results about proper LaFR on join graphs, in addition to those in Section 5. In particular, we determine when and where LaFR can occur on a join graph.
Theorem 11.1. Let Z be a join graph on n ≥ 3 vertices. Suppose Z admits LaFR at time τ . Then τ is an integer multiple of 2π/n.
Proof. Let L be the Laplacian matrix of Z, and L the Laplacian matrix of Z. Recall from the proof of Theorem 5.1 that Assume, for a contradiction, that τ is not an integer multiple of 2π/n. Then in the above formula, the coefficient before J is non-zero. Since X has at least three vertices, this can only happen if X is connected. However, that contradicts the fact that X is a join graph.
We saw in Theorem 5.1 that, under a mild conditon, LaFR is preserved by taking the complement of the graph. As a direct consequence of Theorem 11.1, we can drop this condition when the graph is a join.
Corollary 11.2. Let Z be a connected join graph. Then Z admits LaFR between a and b if and only if Z admits LaFR between a and b at the same time.
The following result determines where LaFR can occur on a join graph. Our theory enables us to build infinite families of join graphs that admit proper LaFR. Unlike those constructed in Theorem 5.3, these graphs do not have to be double cones. To start, we cite a spectral result from [1].
Lemma 11.4. [1] Let X and Y be two graphs on m and n vertices, respectively. Then m + n is an eigenvalue of X + Y with an eigenvector being n 1 m −m 1 n .
We remark an immediate consequence of the structure of the eigenvectors of a join.
Corollary 11.5. Let X + Y be a join graph on ℓ vertices, and let a and b be two Laplacian strongly cospectral vertices.
(i) If a and b both lie in X or both lie in Y , then ℓ ∈ Φ + ab .
(ii) If a lies in X and b lies in Y , then ℓ ∈ Φ − ab .
We now show that any graph with proper LaFR at a special time yields an infinite family of graphs with LaFR.
Theorem 11.6. Let X be a graph on m ≥ 3 vertices. Suppose for some factor g of m, proper LaFR occurs on X bewteen a and b at time 2π/g. Let Y be any graph on n vertices such that g divides n. Then the join graph X + Y admits proper LaFR between a and b at time 2π/g.

Proof.
Since g divides m, by Theorem 5.1, there is proper LaFR at time 2π/g between a and b on X, and thus on X ∪ Y . As g also divides n, applying Theorem 5.1 yields the result.
The rest of this section is devoted to LaFR on double cones. In Theorem 5.3, we showed that every double cone on n ≥ 4 vertices have proper LaFR at time 2π/n. Here, we prove that the converse is also true. Our result uses the following simple observation.
Lemma 11.7. Let X be a graph on n vertices. If n is an eigenvalue of L(X), then X is a join graph.
Proof. By Theorem 2.2, X has at least two components.
With this, we are able to prove the converse of Theorem 5.3.
Theorem 11.8. Let Z be a graph on n ≥ 3 vertices that admit proper LaFR at time 2π/n. Then Z is a double cone.
By Theorem 2.2, the eigenvalues L(Z) are no greater than n, and as 0 ∈ Φ + ab , we must have g ≤ n. On the other hand, Theorem 7.6 says that proper LaFR must occur at times that are integer multiples of 2π/g. Hence g = n. Now, as Φ + ab contains at least two elements, it can only be that Φ + ab = {0, n}. Therefore, Z is a join graph.
By Corollary 11.3, there is an induced subgraph X of Z containing a and b such that (i) Z = X + Y for some induced subgraph Y of Z, and (ii) X admits proper LaFR between a and b at time 2π/n. We claim that X has no other vertices than a and b. Suppose otherwise, and let m be the number of vertices in X. Since m ≥ 3, by Theorem 7.6, the earliest time when proper LaFR could occur on X is 2π/m. However, as m < n, this contradicts the fact that X admits proper LaFR at time 2π/n. Therefore, m = 2 and Z is indeed a double cone.
We reach the same conclusion if the graph with proper LaFR has an prime number of vertices.
Theorem 11.9. Let p be an odd prime. Let X be a graph on p vertices. Suppose proper LaFR occurs on X. Then X is a double cone.
Proof. Suppose X admits proper LaFR between vertices a and b. Let q be the number of spanning trees of X. By Lemma 8.2, q is divisible by the product of elements in Φ 0 a . On the other hand, the Matrix-Tree Theorem states that pq equals the product of non-zero eigenvalues of L(X). Hence p divides r Φ a \{0}.
As p is a prime, it lies in Φ a and in particular, it is an eigenvalue of L(X). Therefore, X is a join graph. By Theorem 7.6, the earliest time when proper LaFR occurs is 2π/g, where g = gcd µ r − µ s : µ r , µ s ∈ Φ + ab or µ r , µ s ∈ Φ − ab . From Corollary 11.5 we see that p ∈ Φ + ab , and so g divides p. However g = 1. It follows that g = p, and by Theorem 11.8, X is a double cone.
We noticed, from the data on small graphs, that most examples with proper LaFR are double cones. This is partially explained by the above two results.

Polygamy of proper fractional revival
It is known that perfect state transfer, with respect to either the Laplacian matrix or the adjacency matrix, is monogamous: if a graph admits perfect state transfer from a to b and from a to c, then b = c. In contrast, we show that LaFR can be polygamous: a vertex may be paired with two different vertices for proper LaFR. The following trick enables us to construct an infinite family of examples.
Lemma 12.1. Let X and Y be two graphs on at least three vertices. Suppose X admits proper LaFR from a to b at time 2π/g, and Y admits proper LaFR from c to d at time 2π/h, where g and h are the gcds defined in Theorem 7.6. Let G = gcd Φ a , H = gcd Φ c .
Further assume that G is not divisible by gcd(g, H), and H is not divisible by gcd(h, G). Then X Y admits LaFR from (a, c) to (b, c), from (a, d) to (b, d), from (a, c) to (a, d), and from (b, c) to (b, d).
Proof. By Theorem 7.6 and Theorem 8.3, the following phenomena occur.
(i) At time 2π/ gcd(g, H), X has proper LaFR between a and b, and Y is Laplacian periodic at c and d.
(ii) At time 2π/ gcd(h, G), X is Laplacian periodic at a and b, and Y has proper LaFR between c and d.
The result now follows from Theorem 4.1.
We use this lemma to build an infinite family of graphs on which LaFR is polygamous. Our construction involves two types of distance regular graphs: Hadamard graphs, and distance regular double covers of complete graphs. The eigenvalues of both types of graphs can be found in the table of [5,Sec 3]. Since these graphs are regular, fractional revival with respect to the Laplacian matrix is equivalent to fractional revival with respect to the adjacency matrix.
Given an n×n Hadamard matrix H, we define 4n symbols r + i , r − i , c + i , c − i , where i = 1, 2, · · · , n. The Hadamard graph is a graph with these symbols as vertices, such that r ± i is adjacent to r ± j if H ij = 1, and r ± i is adjacent to r ∓ j if H ij = −1. It is shown in [5] that the Hadamard graph has proper fractional revival between antipodal vertex a and b with Φ + ab = {0, n 2 , 2n 2 }, Φ − ab = {n 2 − n, n 2 + n}.
In [5], the author also characterized fractional revival on distance regular double cover of K m . If m = 4p 2 for some odd prime p and δ = 2, then the double cover has proper LaFR between the antipodal vertices c and d with It is known that there exist distance regular antipodal covers of K 36 with δ = 2. Thus we have the following.
Theorem 12.2. Let n = 6q for some odd positive integer q. Let X be a Hadamard graph on 4n 2 vertices. Let Y be a distance regular double cover of K 36 with δ = 2. If a and b are antipodal vertices in X, and c and d are antipodal vertices in Y , then the following occur.
(i) X Y admits LaPST from (a, c) to (b, c) and from (a, d) to (b, d) at time π/2.
(ii) X Y admits proper LaFR from (a, c) to (a, d) and from (b, c) to (b, d) at time π/3.
Thus gcd(g, H) does not divide G, and gcd(h, G) does not divide H. Now apply Lemma 12.1.
We remark that, as these graphs are regular, they are also the first infinite family of unweighted graphs that admit polygamous adjacency fractional revival. This answers an open question in [6].