Quantum walks and graph operations
Let $U_X(t)$ be the transition matrix of a quantum walk on a graph $X$ relative to its adjacency matrix $A$ or the Laplacian matrix $L$. This paper investigates the behavior of quantum walks under Cartesian products, joins, and graph complements. We have two main goals. First, we characterize the conditions such that peak state transfer and pretty good state transfer are preserved under these operations, allowing us to construct new families of graphs admitting these properties. Our second goal is to analyze the relationship between the quantum walks on a graph and its complement. We provide bounds for $f_{u,v}(t)=\big|U_{X^c}(t)_{u,v}-e^{itδ}U_X(-t)_{u,v}\big|$ and $g_{u,v}(t)=\big||U_X(t)_{u,v}|-|U_{X^c}(t)_{u,v}|\big|$, where $δ=-1$ when dealing with $A$ and $δ=n$ otherwise. Note that $f_{u,v}(t)$ and $g_{u,v}(t)$ both measure the difference between the behavior of quantum state transfer between vertices $u$ and $v$ in a graph and its complement. If $X$ is regular or $M=L$, then $f_{u,v}(t)$ is bounded above by $\frac{2}{|V(X)|}$. If $X$ is non-regular and $M=A$, then we utilize the main eigenvalues of a graph to obtain an upper bound for $f_{u,v}(t)$ which depends only on $A$. We also use the bounding matrix of the graph to give bounds for the Nordhaus-Gaddum type relations $|U_X(t)_{u,v}|+|U_{X^c}(t)_{u,v}|$ and $|U_X(t)_{u,v}|\cdot |U_{X^c}(t)_{u,v}|$. Finally, we demonstrate that most of our bounds are sharp for certain families of graphs.