2025/08/25 by Kalita, Akash, Bhattacharjya, Bikash
#05C50 #11A07 #15A16 #81P45 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2508.18068
The unitary Cayley graph, denoted Xn, is the graph with vertex set ℤn such that two distinct vertices a and b are adjacent if a-b=u for some u with 1 ≤ u ≤ n-1 and gcd(u,n) = 1. The quadratic unitary Cayley graph, denoted Gn, is the graph with vertex set ℤn such that two distinct vertices a and b are adjacent if a-b=u2 or a-b=-u2 for some u with 1 ≤ u ≤ n-1 and gcd(u,n) = 1. In this paper, we classify all Xn admitting pretty good fractional. We also classify all Xn that admit fractional revival. It turns out that Xn admits fractional revival if and only if it admits pretty good fractional revival. Further, we classify all Gn admitting periodicity. As a consequence, we obtain all Gn admitting perfect state transfer. We also classify Gn admitting pretty good state transfer, pretty good fractional revival and fractional revival.