Sedentariness in Cayley graphs
Abstract
Let A be the adjacency matrix of a finite graph Γ. We say that vertex u of Γ is sedentary if for some constant C>0, |exp(itA)u,u|≥ C for all t∈ℝ. In this paper, we investigate vertex sedentariness in walk-regular graphs with a focus on Cayley graphs and, more generally, vertex-transitive graphs. In particular, we define the necklace polynomial of an integral graph and use this notion to characterize sedentariness in integral walk-regular graphs. As an application, we use basic tools from algebraic number theory to show that an integral Cayley graph on a group of odd order is sedentary and provide a lower bound on its sedentariness. We also give sufficient conditions such that a unitary Cayley graph over a finite ring is sedentary. Furthermore, we characterize the existence of twin vertices in Cayley graphs and vertex-transitive graphs, and completely determine the occurrence of sedentariness, perfect state transfer and pretty good state transfer in vertex-transitive graphs with twins. Finally, we prove that if Γ(G,S) is a nonempty, integral, normal Cayley graph, then for any prime p with gcd(p, |G|)=1, there is at least one eigenvalue of Γ(G,S) which is not a multiple of p. This generalizes a result of Arnadottir and Godsil, which states that an integral normal Cayley graph of odd order possesses an odd eigenvalue.
Related
BibTeX
@misc{meagher2026sedentariness,
title = {{Sedentariness in Cayley graphs}},
author = {Karen Meagher and Hermie Monterde and Tung T. Nguyen},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00567},
eprint = {2611.00567},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00567},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sedentariness in Cayley graphs
AU - Karen Meagher
AU - Hermie Monterde
AU - Tung T. Nguyen
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00567
UR - https://arxiv.org/abs/2611.00567
AB - Let A be the adjacency matrix of a finite graph Γ. We say that vertex u of Γ is sedentary if for some constant C>0, |exp(itA)_u,u|≥ C for all t∈ℝ. In this paper, we investigate vertex sedentariness in walk-regular graphs with a focus on Cayley graphs and, more generally, vertex-transitive graphs. In particular, we define the necklace polynomial of an integral graph and use this notion to characterize sedentariness in integral walk-regular graphs. As an application, we use basic tools from algebraic number theory to show that an integral Cayley graph on a group of odd order is sedentary and provide a lower bound on its sedentariness. We also give sufficient conditions such that a unitary Cayley graph over a finite ring is sedentary. Furthermore, we characterize the existence of twin vertices in Cayley graphs and vertex-transitive graphs, and completely determine the occurrence of sedentariness, perfect state transfer and pretty good state transfer in vertex-transitive graphs with twins. Finally, we prove that if Γ(G,S) is a nonempty, integral, normal Cayley graph, then for any prime p with gcd(p, |G|)=1, there is at least one eigenvalue of Γ(G,S) which is not a multiple of p. This generalizes a result of Arnadottir and Godsil, which states that an integral normal Cayley graph of odd order possesses an odd eigenvalue.
ER -