State Transfer on Unitary Cayley Graphs and Quadratic Unitary Cayley Graphs

The unitary Cayley graph, denoted $X_n$, is the graph with vertex set ${\mathbb{Z}}_n$ such that two distinct vertices $a$ and $b$ are adjacent if $a-b=u$ for some $u$ with $1\le u\le n-1$ and $\gcd (u,n)=1$. The quadratic unitary Cayley graph, denoted $G_n$, is the graph with vertex set ${\mathbb{Z}}_n$ such that two distinct vertices $a$ and $b$ are adjacent if $a-b=u^2$ or $a-b=-u^2$ for some $u$ with $1\le u\le n-1$ and $\gcd (u,n)=1$. In this paper, we classify all $X_n$ admitting pretty good fractional. We also classify all $X_n$ that admit fractional revival. It turns out that $X_n$ admits fractional revival if and only if it admits pretty good fractional revival. Further, we classify all $G_n$ admitting periodicity. As a consequence, we obtain all $G_n$ admitting perfect state transfer. We also classify $G_n$ admitting pretty good state transfer, pretty good fractional revival and fractional revival.