Quantum State Transfer on Neighborhood Corona of Two Graphs

Given two graphs $G_1$ of order $n_1$ and $G_2$, the neighborhood corona of $G_1$ and $G_2$, denoted by $G_1\mathrm{★}{G}_2$, is the graph obtained by taking one copy of $G_1$ and taking $n_1$ copies of $G_2$, in the meanwhile, linking all the neighbors of the $i$-th vertex of $G_1$ with all vertices of the $i$-th copy of $G_2$. In our work, we give some conditions that $G_1\mathrm{★}{G}_2$ is not periodic. Furthermore, we demonstrate some sufficient conditions for $G_1\mathrm{★}{G}_2$ having no perfect state transfer. Some examples are provided to explain our results. In addition, for the reason that the graph admitting perfect state transfer is rare, we also consider pretty good state transfer on neighborhood corona of two graphs. We show some sufficient conditions for $G_1\mathrm{★}{G}_2$ admitting pretty good state transfer.