Periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs

The quadratic unitary Cayley graph Gₙ has vertex set Zₙ: =\0,1, ,n-1\, where two vertices u and v are adjacent if and only if $u - v$ or $v-u$ is a square of some units in Zₙ. This paper explores the periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs. We determine all periodic quadratic unitary Cayley graphs. From our results, it follows that there are infinitely many integral as well as non-integral graphs that are periodic. Additionally, we also determine the values of n for which the quadratic unitary Cayley graph Gₙ exhibits perfect state transfer.