More Circulant Graphs exhibiting Pretty Good State Transfer

Abstract

The transition matrix of a graph $G$ corresponding to the adjacency matrix $A$ is defined by $H(t):=\exp \left(-itA\right),$ where $t\in \mathbb{R}$. The graph is said to exhibit pretty good state transfer between a pair of vertices $u$ and $v$ if there exists a sequence $\left\{t_k\right\}$ of real numbers such that $\underset{k\to \infty }{\lim }H(t_k){\mathbf{e}}_u=\gamma {\mathbf{e}}_v$, where $\gamma$ is a complex number of unit modulus. We classify some circulant graphs exhibiting or not exhibiting pretty good state transfer. This generalize several pre-existing results on circulant graphs admitting pretty good state transfer.