trureturing — Discovering truth

bibkey: hanwangyu2025gluing
authors: X. B. Han; S. Wang; C. Yu
year: 2025
title: "PageRank of Gluing Networks and Corresponding Markov Chains"
doi: 10.3390/math13132080
url: https://www.mdpi.com/2227-7390/13/13/2080
claim: "Glue Markov chains M_i = (S_i, A_i) to a Markov chain M = (S, A), then all the eigenvalues of A lie in the convex hull of the eigenvalues of all A_i on the complex plane."
strata_touched:
  - D5/S3/Quantum/Matrix/HanWangYuGluingSpectrumRefutation
license: citation-only
triage: anchor

Verified locator

DOI: 10.3390/math13132080

Source: https://www.mdpi.com/2227-7390/13/13/2080

Publisher PDF: https://mdpi-res.com/d_attachment/mathematics/mathematics-13-02080/article_deploy/mathematics-13-02080.pdf

Source statements

Definition 8, p. 10:

For integers i = 1, 2, ⋯, k, let M_i = (S_i, A_i) be k Markov chains, and v_0^i be a fixed point in S_i. Assume θ = (θ_1, θ_2, ⋯, θ_k), 0 < θ_i < 1, and Σ_{i=1}^k θ_i = 1.

The θ-gluing of M_i along v_0^i is the following Markov chain M = (S, A).

The state set is the disjoint union of the state sets with their selected roots identified. The transition rule is

$$a_{vw}=\{\begin{array}{ll}{a}_{vw}^i& v\in S_i,w\in S_i\setminus \{v_0\},\\ {\theta }_i{a}_{vw}^i& v\in S_i\setminus \{v_0\},w=v_0,\\ {\sum }_i{\theta }_i{a}_{v_0{v}_0}^i& v=w=v_0,\\ 0& \text{otherwise}.\end{array}$$

Conjecture 1, §4.3, p. 17:

Glue Markov chains M_i = (S_i, A_i) to a Markov chain M = (S, A), then all the eigenvalues of A lie in the convex hull of the eigenvalues of all A_i on the complex plane.

Indices in the formal statement run through Fin k. Each input chain has state set Fin (n + 1) with selected root 0; matrix entry (v,w) gives the probability from w to v. The glued state set is Option (Σ i, Fin (M i).n), where none is the common root and some ⟨i,v⟩ is state v.succ of chain i. Eigenvalues are spectrum ℂ of the entrywise complexification, and the convex hull is over ℝ in ℂ.

Counterexample

Two identical copies of

$$B=\frac{1}{54}\left(\begin{array}{ccc}43& 9& 2\\ 2& 36& 16\\ 9& 9& 36\end{array}\right)$$

with equal weights have a glued eigenvalue $4/9$. The input characteristic polynomial is $(\lambda -1)(\lambda -17/27)(\lambda -1/2)$, so every input eigenvalue has real part at least $1/2$. The vector $(0,-4,3,4,-3)$ is a nonzero glued eigenvector for $4/9$. Its root coordinate is zero and the two private blocks cancel in the root row. These facts yield the repository-derived refutation result : ¬ claim.

Literature boundary

The preregistration at https://github.com/the-omega-institute/trureturing/issues/19143 records the source and the literature screen. Completeness is ASSUMED-UNVERIFIED: Rymkhanova et al. (2026), Schwenk (1974), Laffey–Šmigoc and Belardo–Li Marzi–Simić (2010) were not accessed. General families, similarity decompositions and reversibility conditions are not proved by this module.