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

2024/08/16 by Koushik Bhakta, Bikash Bhattacharjya, Bhakta, Koushik +1 · 4 citations
Mathematics · #05C50 #81Q99 #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.2408.08715

openalex publication_date 2024/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The quadratic unitary Cayley graph G_ℤn has vertex set ℤ_n: =\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 ℤ_n. 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 G_ℤn exhibits perfect state transfer.

Cited by

• Subspace State Transfer in Coined Quantum Walks
• State Transfer on Unitary Cayley Graphs and Quadratic Unitary 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

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• State transfer in Grover walks on unitary and quadratic unitary Cayley graphs over finite commutative rings
• State Transfer on Unitary Cayley Graphs and Quadratic Unitary Cayley Graphs
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