This record contains a short preliminary note (TeX source and PDF), Python code and its output, on the question posed in Section 11 of H. Monterde, H. Pal and S. Kirkland, "Laplacian quantum walks on blow-up graphs" (Linear Algebra Appl., 2025, doi:10.1016/j.laa.2025.10.001; numbering of arXiv:2504.11585v1): whether their Theorem 10 (Laplacian perfect state transfer after adding a matching inside the copies of a vertex in the blow-up of n copies of a graph, n divisible by 4) holds for the adjacency matrix. The same question is Problem 10 in H. Monterde's PhD thesis "Quantum pure state transfer" (University of Manitoba, 2025). For matchings whose edges join twin vertices of the blow-up and even n, the note proves that adjacency perfect state transfer between the ends a, b of an added edge forces the matching to cover every neighbour of a, and gives parity conditions on the uncovered vertices. Consequently a nonempty matching inside the copies of a single non-isolated vertex never gives adjacency perfect state transfer, for every graph and every even n. These results, and exact computations on small graphs supporting a conjecture that all sets of copies must be perfectly matched (finite computations only), are the parts claimed as new; a literature search in October 2026 found no earlier treatment of them, which does not establish that none exists. When every set of copies carries a perfect matching (a lexicographic product G[(n/2)K_2]), the note shows that for n divisible by 4 adjacency perfect state transfer between the ends of an added edge occurs if and only if the adjacency eigenvalue support of the vertex is integral, at time pi/2. This is not claimed as new: since G[(n/2)K_2] is isomorphic to Z[K_2], where Z is the blow-up of n/2 copies of G, it follows in a few lines from Proposition 6.3 of A. S. Árnadóttir and K. Guo (arXiv:2608.20476), and its sufficiency for integral graphs is a special case of a lemma of Ge, Greenberg, Perez and Tamon (arXiv:1009.1340). The criterion used for perfect state transfer between adjacent twins is a special case of a theorem of Kirkland, Monterde and Plosker (arXiv:2201.02720). When the matching lies inside the sets of copies, the graphs are edge-perturbed blow-ups in the sense of Monterde and Pal (arXiv:2505.07982v5, Section 9); their Theorem 20 concerns states orthogonal to the all-ones vector of a set of copies and does not cover vertex states. This is a personal hobby/independent research record in which AI was used for exploration, drafting proof candidates, checking and writing; it has not been peer-reviewed by human experts.
Let X be a weighted graph and M be its adjacency, Laplacian or signless Laplacian matrix. In a continuous quantum walk on X, local ε-uniform mixing occurs at vertex u if the uth column of the matrix U(t)=eitM can be made arbitrarily close to a vector whose all entries have equal magnitude. Using the spectral and combinatorial properties of X, we derive necessary conditions for local ε-uniform mixing to occur in X. This includes an inequality involving all entries of each eigenvector of M, as well as an upper bound on the degree of vertex u when M is the Laplacian or signless Laplacian matrix. We use these necessary conditions to rule out local ε-uniform mixing in numerous classes of graphs, most of which are non-regular. We also show that almost all planar graphs (resp., trees) contain a vertex that does not admit local ε-uniform mixing for any assignment of edge weights. Furthermore, we prove if X has n vertices and admits local ε-uniform mixing at a vertex contained in a subgraph with a twin, then the number of vertices of this twin subgraph must be at least n. In particular, we establish that a graph on n≥5 vertices does not admit local ε-uniform mixing at a vertex with a twin.