Grover walks on unitary Cayley graphs and integral regular graphs

2024/05/02 by Bhakta, Koushik, Bhattacharjya, Bikash · 4 citations
#05C50 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2405.01020

Abstract

The unitary Cayley graph has vertex set \0,1, \hdots ,n-1\, where two vertices u and v are adjacent if gcd(u - v, n) = 1. In this paper, we study periodicity and perfect state transfer of Grover walks on the unitary Cayley graphs. We characterize all periodic unitary Cayley graphs. We prove that periodicity is a necessary condition for occurrence of perfect state transfer on a vertex-transitive graph. Also, we provide a necessary and sufficient condition for the occurrence of perfect state transfer on circulant graphs. Using these, we prove that only four graphs in the class of unitary Cayley graphs exhibit perfect state transfer. Also, we provide a spectral characterization of the periodicity of Grover walks on integral regular graphs.

Cited by

• Subspace State Transfer in Coined Quantum Walks
• Pretty good state transfer in Grover walks on abelian Cayley graphs
• Perfect state transfer in Grover walks on association schemes and distance-regular graphs
• Strongly regular and strongly walk-regular graphs that admit perfect state transfer

Related

• Periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs
• State transfer in Grover walks on unitary and quadratic unitary Cayley graphs over finite commutative rings
• Perfect state transfer in Grover walks on normal Cayley graphs
• Pretty good state transfer in Grover walks on abelian Cayley graphs
• Strongly regular and strongly walk-regular graphs that admit perfect state transfer