bibkey: monterdepalkirkland2025blowup
authors: Hermie Monterde; Hiranmoy Pal; Steve Kirkland
year: 2025
title: "Laplacian quantum walks on blow-up graphs"
doi: 10.1016/j.laa.2025.10.001
url: https://arxiv.org/abs/2504.11585v1
claim: "The paper studies Laplacian quantum walks on the n-fold blow-up of a graph, whose clones of a vertex are pairwise false twins. It characterizes periodicity, strong cospectrality and Laplacian perfect and pretty good state transfer in blow-ups, and constructs blow-ups with Laplacian perfect state transfer. For n divisible by four and a vertex with integral Laplacian eigenvalue support, adding a matching among the clones of that vertex gives Laplacian perfect state transfer at pi/2 between the ends of every inserted edge. It asks whether this holds for the adjacency matrix."
strata_touched:
  - D5/S3/Quantum/Dynamics/BlowUpMatchingAdjacencyTransfer
license: citation-only
triage: anchor

Monterde, Pal and Kirkland, Laplacian quantum walks on blow-up graphs

H. Monterde, H. Pal and S. Kirkland, arXiv:2504.11585v1 [math.CO]; Linear Algebra and its Applications (2025). Quotations are from the v1 TeX source, with the source’s macros for the blow-up and the integers expanded.

The $n$-fold blow-up ${\uparrow }^{n}G$ has vertices $(j,v)$, $j\in {\mathbb{Z}}_n$; the set $T_v=\{(j,v)\mid j\in {\mathbb{Z}}_n\}$ consists of pairwise false twins. The theorem labelled pt1 (Section 10, Edge perturbation):

Let $n\equiv 0$ (mod 4) and $v$ be a vertex in $G$ with ${\sigma }_v(G)\subset \mathbb{Z}.$ Then the addition of a matching in $T_v$ results in LPST in ${\uparrow }^{n}G$ at $\frac{\pi }{2}$ between the end vertices of every edge inserted. Moreover, the resulting graph is periodic at all vertices in $T_v$ that are not incident to the newly added edges with period $\frac{\pi }{2}.$

The last question of Section 11 (Future work):

Lastly, we ask: does Theorem \ref{pt1} hold for the adjacency case? That is, if $n\equiv 0$ (mod 4), will the addition of an appropriate matching in ${\uparrow }^{n}G$ induce adjacency LPST?

The same question is Problem 10 of the first author’s thesis, Quantum pure state transfer (University of Manitoba, 2025), printed page 163.

Verified locator

Source: https://arxiv.org/abs/2504.11585v1

• DOI: https://doi.org/10.1016/j.laa.2025.10.001 (Linear Algebra and its Applications, 2025; title, author list and journal from Crossref).
• URL: https://arxiv.org/abs/2504.11585v1. The e-print is a gzipped tar archive; its member main.tex has 824 lines and md5 b21c68f0d57c75bbf8eb7530da0c4d04. Locations:
  • Section 10 (Edge perturbation), l. 662; Theorem pt1, l. 666–669;
  • Section 11 (Future work), l. 807; the question, l. 814.
• Thesis: https://mspace.lib.umanitoba.ca/bitstreams/5459043e-de2b-4617-b323-694eca552e14/download (Problem 10, PDF page 173, printed page 163).
• The arXiv version (v1, 15 April 2025) is from the arXiv metadata; the record lists no other version.