A complete classification of perfect state transfer in Grover walks on abelian Cayley graphs
Year: 2026
Type: preprint
Abstract: We completely classify connected Cayley graphs over finite abelian groups that exhibit perfect state transfer in Grover walks. First, we show that a Cayley graph over any finite group exhibiting perfect state transfer has valency $1$, $2$, or $4$. For valency $4$, we prove that an abelian Cayley graph $\operatorname{Cay}(Γ,S)$ exhibits perfect state transfer if and only if $S=\{a,-a,a+h,-a+h\}$, where $h$ has order $2$, and $4\nmid |Γ|/2$. Combining this characterization with the valency-one and valency-two cases, we explicitly determine all pairs $(Γ,S)$ for which $\operatorname{Cay}(Γ,S)$ exhibits perfect state transfer. Our results extend the classification of circulant graphs of valency at most four obtained by Kubota and Yoshino [Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks, \emph{J. Comb. Theory Ser. A.}, 216 (2025) 106064] to Cayley graphs over arbitrary finite abelian groups and to all valencies.
Source: arXiv (Cornell University)
Authors: Bhakta, Koushik, Bhattacharjya, Bikash
Institution: Indian Institute of Technology Guwahati
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222727177