Strong Cospectrality and Twin Vertices in Weighted Graphs

We explore algebraic and spectral properties of weighted graphs containing twin vertices that are useful in quantum state transfer. We extend the notion of adjacency strong cospectrality to arbitrary Hermitian matrices, with focus on the generalized adjacency matrix and the generalized normalized adjacency matrix. We then determine necessary and sufficient conditions such that a pair of twin vertices in a weighted graph exhibits strong cospectrality with respect to the above-mentioned matrices. We also generalize known results about equitable and almost equitable partitions, and use these to determine which joins of the form $X\vee H$, where $X$ is either the complete or empty graph, exhibit strong cospectrality.Comment: 25 pages, 6 figure

Sedentariness in quantum walks

We present a relaxation of the concept of a sedentary family of graphs introduced by Godsil [Linear Algebra Appl. 614:356-375, 2021] and provide sufficient conditions for a given vertex in a graph to exhibit sedentariness. We show that a vertex with at least two twins (vertices that share the same neighbours) is sedentary. We also prove that there are infinitely many graphs containing strongly cospectral vertices that are sedentary, which reveals that, even though strong cospectrality is a necessary condition for pretty good state transfer, there are strongly cospectral vertices which resist high probability state transfer to other vertices. Moreover, we derive results about sedentariness in products of graphs which allow us to construct new sedentary families, such as Cartesian powers of complete graphs and stars.Comment: 26 pages, 3 figure

New results in vertex sedentariness

A vertex in a graph is said to be sedentary if a quantum state assigned on that vertex tends to stay on that vertex. Under mild conditions, we show that the direct product and join operations preserve vertex sedentariness. We also completely characterize sedentariness in blow-up graphs. These results allow us to construct new infinite families of graphs with sedentary vertices. We prove that a vertex with a twin is either sedentary or admits pretty good state transfer. Moreover, we give a complete characterization of twin vertices that are sedentary, and provide sharp bounds on their sedentariness. As an application, we determine the conditions in which perfect state transfer, pretty good state transfer and sedentariness occur in complete bipartite graphs and threshold graphs of any order.Comment: 19 pages, 1 figur

Quantum walks on join graphs

The join $X\vee Y$ of two graphs $X$ and $Y$ is the graph obtained by joining each vertex of $X$ to each vertex of $Y$. We explore the behaviour of a continuous quantum walk on a weighted join graph having the adjacency matrix or Laplacian matrix as its associated Hamiltonian. We characterize strong cospectrality, periodicity and perfect state transfer (PST) in a join graph. We also determine conditions in which strong cospectrality, periodicity and PST are preserved in the join. Under certain conditions, we show that there are graphs with no PST that exhibits PST when joined by another graph. This suggests that the join operation is promising in producing new graphs with PST. Moreover, for a periodic vertex in $X$ and $X\vee Y$, we give an expression that relates its minimum periods in $X$ and $X\vee Y$. While the join operation need not preserve periodicity and PST, we show that $\big| |U_M(X\vee Y,t)_{u,v}|-|U_M(X,t)_{u,v}| \big|\leq \frac{2}{|V(X)|}$ for all vertices $u$ and $v$ of $X$, where $U_M(X\vee Y,t)$ and $U_M(X,t)$ denote the transition matrices of $X\vee Y$ and $X$ respectively relative to either the adjacency or Laplacian matrix. We demonstrate that the bound $\frac{2}{|V(X)|}$ is tight for infinite families of graphs.Comment: 29 page

Quantum walks on blow-up graphs

A blow-up of $n$ copies of a graph $G$ is the graph $\overset{n}\uplus~G$ obtained by replacing every vertex of $G$ by an independent set of size $n$, where the copies of vertices in $G$ are adjacent in the blow-up if and only if the vertices adjacent in $G$. Our goal is to investigate the existence of quantum state transfer on a blow-up graph $\overset{n}\uplus~G$, where the adjacency matrix is taken to be the time-independent Hamiltonian of the quantum system represented by $\overset{n}\uplus~G$. In particular, we establish necessary and sufficient conditions for vertices in a blow-up graph to exhibit strong cospectrality and various types of high probability quantum transport, such as periodicity, perfect state transfer (PST) and pretty good state transfer (PGST). It turns out, if $\overset{n}\uplus~G$ admits PST or PGST, then one must have $n=2.$ Moreover, if $G$ has an invertible adjacency matrix, then we show that every vertex in $\overset{2}\uplus~G$ pairs up with a unique vertex to exhibit strong cospectrality. We then apply our results to determine infinite families of graphs whose blow-ups admit PST and PGST.Comment: 14 pages, 2 figure

Quantum state transfer between twins in weighted graphs

Twin vertices are vertices of a graph that have the same neighbours and, in the case of weighted graphs with possible loops, corresponding edges have the same edge weight. We explore the role of twin vertices in quantum state transfer. In particular, we provide a characterization of perfect state transfer, pretty good state transfer and fractional revival between twin vertices in a weighted graph with respect to its adjacency and Laplacian matrices.Comment: 25 page

On the Sum of Strictly k-zero Matrices

Let k be an integer such that k ≥ 2. An n-by-n matrix A is said to be strictly k-zero if Ak = 0 and Am ≠ 0 for all positive integers m with m < k. Suppose A is an n-by-n matrix over a field with at least three elements. We show that, if A is a nonscalar matrix with zero trace, then (i) A is a sum of four strictly k-zero matrices for all k ∈{2,..., n}; and (ii) A is a sum of three strictly k-zero matrices for some k ∈{2,..., n}. We prove that, if A is a scalar matrix with zero trace, then A is a sum of five strictly k-zero matrices for all k ∈{2,..., n}. We also determine the least positive integer m, such that every square complex matrix A with zero trace is a sum of m strictly k-zero matrices for all k ∈{2,..., n}