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 $\left\lbrace t_k\right\rbrace$ of real numbers such that $\lim\limits_{k\rightarrow\infty} H(t_k) {\bf e}_u=\gamma {\bf e}_v$, 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.