bibkey: monterde2023sedentariness authors: Hermie Monterde year: 2023 title: “New results in vertex sedentariness” doi: 10.1016/j.disc.2025.114959 url: https://arxiv.org/abs/2401.00362v1 claim: “We conjecture that u is sharply (1/n)-sedentary in P_n’ for all odd n >= 5 (Example 18; P_n’ is the path P_n with a vertex v = n + 1 added as a non-adjacent twin of the end vertex u = 1, and u is sharply C-sedentary when inf_{t>0} |U(t)_{u,u}| = C for U(t) = e^{itA}).” strata_touched:

• D5/S3/Quantum/Dynamics/PathTwinSharpSedentariness license: citation-only triage: anchor

New results in vertex sedentariness

H. Monterde, arXiv:2401.00362 (v1 2023-12-31, the only version); Discrete Math. 349(4) (2026) 114959. Subjects: math.CO, quant-ph.

The paper studies continuous-time quantum walks U(t) = e^{itM} for real symmetric graph matrices M. A vertex u is C-sedentary if inf_{t>0} |U(t)_{u,u}| ≥ C for a constant 0 < C ≤ 1, and sharply C-sedentary if equality holds. Example 18 takes M = A, n ≥ 3 odd and the graph P_n' obtained from the path P_n by adding a vertex v = n + 1 that is a non-adjacent twin of the end vertex u = 1. It shows |U(t)_{u,u}| ≥ 1/n for all t, proves sharpness for n = 5 from the linear independence over ℚ of the positive eigenvalues in the support of u (the paper says “nonzero eigenvalues”; they come in pairs ±λ), and states:

We conjecture that $u$ is sharply $(\frac{1}{n})$-sedentary in $P_n^{\prime }$ for all odd $n\ge 5$.

Verified locator

• DOI: https://doi.org/10.1016/j.disc.2025.114959 (CC BY 4.0 per Crossref; the journal text was not retrieved: ScienceDirect HTTP 403, Elsevier API 401).
• URL: https://arxiv.org/abs/2401.00362v1 (source retrieved 2026-10-01): Definition 1 (l. 238–245), Lemma 9 (l. 316–322), Example 18 (l. 465–472).