Quantum walks on join graphs

The graphs $K_{2}$, $C_{4}$ $P_{3}$, and $K_{4}\backslash e$ are well-known examples of small graphs that admit adjacency or Laplacian perfect state transfer. Kirkland et al. noticed that in these small graphs, the vertices involved in perfect state transfer share the same neighbours, an observation that prompted them to examine state transfer between twins [Kirkland2023]. However, it can also be observed that these small graphs are in fact join graphs. This motivates our investigation on quantum walks on graphs built using the join operation.

The first infinite family of join graphs revealed to admit Laplacian perfect state transfer was $K_{n}\backslash e:=O_{2}\vee K_{n-2}$ (the complete graph on $n$ vertices minus an edge) with $n\equiv 0$ (mod 4) [Bose2008]. Motivated by this result, Angeles-Canul et al. gave sufficient conditions for adjacency perfect state transfer to occur between the vertices of $X\in\{O_{2},K_{2}\}$ in the unweighted graph $X\vee Y$, where $Y$ is a regular graph [Angeles-Canul2010]. Such a graph is called a double cone on $Y$, and the vertices of $X\in\{O_{2},K_{2}\}$ are called the apexes of the double cone. They also determined sufficient conditions such that adjacency perfect state transfer in $X$ is preserved under joins of copies of $X$. In a subsequent paper, Angeles-Canul et al. investigated perfect state transfer in weighted join graphs, and found that the apexes of a double cone on a $k$-regular graph admit adjacency perfect state transfer by appropriate choice of weights of an edge between the apexes and/or loops on the apexes [Angeles-Canul2009]. More recently, Kirkland et al. fully characterized unweighted double cones that admit adjacency perfect state transfer between apexes [Kirkland2023]. For the Laplacian case, Alvir et al. showed that the apexes of unweighted $O_{2}\vee Y$ admit perfect state transfer if and only if $|V(Y)|\equiv 2$ (mod 4), while the apexes of unweighted $K_{2}\vee Y$ do not admit perfect state transfer [Alvir2016]. Joins have also been investigated in graphs with well-structured eigenbases [Johnston2017, mclaren2023weak], and in the contexts of fractional revival [Monterde2023a, Chan2021], sedentariness [Monterde2023] and strong cospectrality [Monterde2022].

Despite its widespread presence in literature, state transfer on join graphs remains largely unexplored. In this paper, we provide a systematic study of quantum walks on weighted join graphs having the adjacency and Laplacian matrices as their associated Hamiltonian. In Section 3, we provide known results about transition matrices and eigenvalue supports of vertices in join graphs. In Section 4, we characterize periodic vertices in a join (Theorem 4), and determine the conditions in which periodicity of a vertex in $X$ and $X\vee Y$ are equivalent (Corollaries 7 and 9). We also use the join operation to provide infinite families of graphs that are not integral (resp., Laplacian integral) whereby each member graph contains periodic vertices (Corollary 12). We devote Section 5 to finding a relationship between the minimum periods $\rho_{X}$ and $\rho_{X\vee Y}$ of a vertex that is periodic in both $X$ and $X\vee Y$, resp. (Theorems 14 and 15). We find that $\rho_{X\vee Y}$ is a rational multiple of $\rho_{X}$, and if we add the extra hypothesis that $X$ is disconnected, then $\rho_{X\vee Y}$ is an integer multiple of $\rho_{X}$. Section 6 provides a characterization of strong cospectrality in joins (Theorem 20). It turns out that strong cospectrality between two vertices of $X$ in $X\vee Y$ requires strong cospectrality in $X$, except for the case when $X$ is an empty graph on two vertices. In fact, for the unweighted case, strong cospectrality in $X$ and $X\vee Y$ are equivalent with a few exceptions (Corollary 20). In Section 7, we characterize perfect state transfer in joins (Theorems 23 and 26). Unlike strong cospectrality, perfect state transfer between two vertices of $X$ in $X\vee Y$ does not require perfect state transfer between them in $X$. This motivates us in Section 8 to determine necessary and sufficient conditions such that perfect state transfer in $X$ is preserved in $X\vee Y$ (Corollaries 28 and 35), and in Section 9 to determine conditions such that perfect state transfer occurs in $X\vee Y$ given that it does not occur in $X$ (Theorems 40 and 42, and Corollary 46). In Section 10, we characterize strong cospectrality and PST in self-joins and iterated join graphs. The latter result generalizes a previously known characterization of perfect state transfer in threshold graphs. While the join operation does not preserve periodicity and PST in general, we show in Section 11 that $\big{|}|U_{M}(X\vee Y,t)_{u,v}|-|U_{M}(X,t)_{u,v}|\big{|}\leq\frac{2}{|V(X)|}$ for all $t$ and for all vertices $u$ and $v$ of $X$ (Corollaries 70 and 72). Consequently, the behaviour of vertices in $X$ in the quantum walk on $X\vee Y$ mimics the behaviour of the quantum walk on $X$ as $|V(X)|\rightarrow\infty$ (Corollary 73). We also show that this upper bound is tight for infinite families of graphs (Examples 75 and 76). Finally, open problems are presented in Section 12.

Throughout, we assume that a graph $X$ is weighted and undirected with possible loops but no multiple edges. We denote the vertex set of $X$ by $V(X)$, and we allow the edges of $X$ to have positive real weights. We say that $X$ is simple if $X$ has no loops, and $X$ is unweighted if all edges of $X$ have weight one. For $u\in V(X)$, we denote the the characteristic vector of $u$ as $\textbf{e}_{u}$, which is a vector with a $1$ on the entry indexed by $u$ and zeros elsewhere. The all-ones vector of order $n$, the zero vector of order $n$, the $m\times n$ all-ones matrix, and the $n\times n$ identity matrix are denoted by $\textbf{1}_{n}$, $\textbf{0}_{n}$, $\textbf{J}_{m,n}$ and $I_{n}$, respectively. If $m=n$, then we write $\textbf{J}_{m,n}$ as $\textbf{J}_{n}$, and if the context is clear, then we simply write these matrices as 1, 0, J and $I$, respectively. We also represent the transpose of the matrix $H$ by $H^{T}$, and the characteristic polynomial of $H$ in the variable $t$ by $\phi(H,t)$. Lastly, we denote the simple unweighted empty, cycle, complete, and path graphs on $n$ vertices as $O_{n}$, $C_{n}$, $K_{n}$, and $P_{n}$, respectively. We also denote the hypercube on $2^{p}$ vertices by $Q_{p}$, and the cocktail party graph on $m$ vertices by $CP(m)$, where $m$ is even.

The adjacency matrix $A(X)$ of $X$ is the matrix defined entrywise as

$$A(X)_{u,v}=\begin{cases}\omega_{u,v},&\text{if $u$ and $v$ are adjacent}\\ 0,&\text{otherwise},\end{cases}$$

where $\omega_{u,v}$ is the weight of the edge $(u,v)$. The degree matrix $D(X)$ of $X$ is the diagonal matrix of vertex degrees of $X$, where $\operatorname{deg}(u)=2\omega_{u,u}+\sum_{j\neq u}\omega_{u,j}$ for each $u\in V(X)$. The Laplacian matrix of $X$ is the matrix $L(X)=D(X)-A(X)$. We use $M(X)$ to denote $A(X)$ or $L(X)$. If the context is clear, then we write $M(X)$, $A(X)$, $L(X)$ and $D(X)$ resp. as $M$, $A$, $L$ and $D$. We say that $X$ is integral (resp., Laplacian integral) if all eigenvalues of $A(X)$ (resp., $L(X)$) are integers. We say that $X$ is weighted $k$-regular if $\operatorname{deg}(u)-\omega_{u,u}$ is a constant $k$ for each $u\in V(X)$, i.e., the row sums of $A(X)$ are a constant $k$. If $X$ is simple, then $X$ is weighted $k$-regular if and only if $D(X)=kI$.

A (continuous-time) quantum walk on $X$ with respect to $H$ is determined by the unitary matrix

$$U_{H}(X,t)=e^{itH}.$$

(1)

The matrix $H$ is called the Hamiltonian of the quantum walk. Typically, $H$ is taken to be $A$ or $L$, but any Hermitian matrix $H$ that respects the adjacencies of $X$ works in general. Since $M\in\{A,L\}$ is real symmetric, $M$ admits a spectral decomposition

$$M=\sum_{j}\lambda_{j}E_{j},$$

(2)

where the $\lambda_{j}$’s are the distinct real eigenvalues of $M$ and each $E_{j}$ is the orthogonal projection matrix onto the eigenspace associated with $\lambda_{j}$. If the eigenvalues are not indexed, then we also denote by $E_{\lambda}$ the orthogonal projection matrix corresponding to the eigenvalue $\lambda$ of $M$. Now, (2) allows us to write (1) as

$$U_{M}(X,t)=\sum_{j}e^{it\lambda_{j}}E_{j}.$$

We say that perfect state transfer (PST) occurs between two vertices $u$ and $v$ if $|U_{M}(X,\tau)_{u,v}|=1$ for some time $\tau$. The minimum $\tau>0$ such that $|U_{M}(X,\tau)_{u,v}|=1$ is called the minimum PST time between $u$ and $v$. We say that $u$ is periodic if $|U_{M}(X,\tau)_{u,u}|=1$ for some time $\tau$. The minimum $\tau>0$ such that $|U_{M}(X,\tau)_{u,u}|=1$ is called the minimum period of $u$. These properties depend on the matrix $M$, so we sometimes say adjacency PST (resp., periodicity) when $M=A$; similar language applies when $M=L$. Further, if $X$ is simple, weighted and $k$-regular, then $U_{L}(X,t)=e^{itL}=e^{it(kI-A)}=e^{itk}e^{-itA}=e^{itk}U_{A}(X,-t)$. Hence, $\left|U_{L}(X,\tau)_{u,v}\right|^{2}=\left|U_{A}(X,\tau)_{u,v}\right|^{2}$ for any $u,v\in V(X)$, so the quantum walks on $A$ and $L$ exhibit the same state transfer properties.

The join $X\vee Y$ of weighted graphs $X$ and $Y$ is the graph obtained by joining every vertex of $X$ with every vertex of $Y$ with an edge of weight one. Unless otherwise stated, we make the following assumptions:

• •

  when dealing with $M=L,$ we take $X$ and $Y$ to be simple;

• •

  when dealing with $M=A,$ we take $X$ and $Y$ to be weighted $k$- and $\ell$-regular resp. (possibly with loops).

Denote the transition matrices of $X\vee Y$ and $X$ by $U_{M}(X\vee Y,t)$ and $U_{M}(X,t)$ respectively. The transition matrix of a join with respect to $L$ and $A$ are known (see [Alvir2016, Fact 8] and [Coutinho2021, Lemma 4.3.1], respectively). Throughout, $m$ and $n$ denote the number of vertices of $X$ and $Y$, respectively.

The eigenvalue support of $u$ with respect to $M$ is the set

$$\sigma_{u}(M)=\{\lambda_{j}:E_{j}\textbf{e}_{u}\neq\textbf{0}\}.$$

The following result determines the elements of the eigenvalue support of a vertex in a join graph. In what follows, $\sigma_{u}(M(X))$ and $\sigma_{u}(M)$ denote the eigenvalue supports of vertex $u$ in $X$ and $X\vee Y$, respectively.

A set $S\subseteq\mathbb{R}$ with at least two elements satisfies the ratio condition if $\frac{\lambda-\theta}{\mu-\eta}\in\mathbb{Q}$ for all $\lambda,\theta,\mu,\eta\in S$ with $\mu\neq\eta$. A vertex $u$ is periodic if and only if $\sigma_{u}(M)$ satisfies the ratio condition [Coutinho2021, Corollary 8.3.1]. Combining this fact with Lemma 3 yields a characterization of periodicity in joins.

The following example illustrates that the converse of the last statement in Theorem 4(1) does not hold.

If $\phi(M(X),t)\in\mathbb{Z}[t]$, then we obtain another characterization of periodicity [Coutinho2021, Corollary 7.6.2].

We now show that the converse of the last statement in Theorem 4(1) holds when $\phi(L(X),t)\in\mathbb{Z}[t]$.

Next, we have the following result concerning the adjacency matrix in relation to Theorem 4(2).

We now state an analogue of Corollary 7, which is an immediate consequence of Corollary 8.

We say that $X$ is periodic if there is a $\tau>0$ such that $U(\tau)=\gamma I$ for some unit $\gamma\in\mathbb{C}$. Equivalently, each $u\in V(X)$ is periodic at time $\tau$. We characterize periodic join graphs under mild conditions.

We close this section by providing infinite families of graphs that are not (Laplacian) integral but contain periodic vertices. The following result is immediate from Lemma 3 and the ratio condition.

Denote the minimum periods of $u$ in $X\vee Y$ and $X$ by $\rho_{X\vee Y}$ and $\rho_{X}$ respectively. Motivated by Corollaries 7 and 9, we ask: if $u\in V(X)$ is periodic in $X$ and $X\vee Y$, then how are $\rho_{X\vee Y}$ and $\rho_{X}$ related? To answer this question, we state a result due to Kirkland et al. [Kirkland2023, Theorem 4].

We now address our above question by finding a constant $c>0$ such that $\rho_{X\vee Y}=c\rho_{X}$.