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Covariance matrices of volume power functionals of random simplicial complexes -- an asymptotic analysis
Mandala von Westenholz
https://arxiv.org/abs/2509.15790 https://

Covariance matrices of volume power functionals of random simplicial complexes -- an asymptotic analysis
This work analyzes and compares the asymptotic properties of the covariance matrices of vectors of volume power functionals of random Vietoris-Rips complexes, as the intensity of the underlying homogeneous Poisson point process grows. Several key results are established which, in particular, generalize well-known facts on random graphs. Findings regarding rank, definiteness, determinant, eigenspaces, and related decompositions are presented within three distinct regimes. Moreover, we derive sto…

Spectral properties of distance Laplacian matrices of complex unit gain graphs
Aniruddha Samanta, Deepshikha
https://arxiv.org/abs/2507.21345 https://arxi…

Spectral properties of distance Laplacian matrices of complex unit gain graphs
A complex unit gain graph ($ \mathbb{T} $-gain graph), $ Φ=(G, φ) $ is a graph where the function $ φ$ assigns a unit complex number to each orientation of an edge of $ G $, and its inverse is assigned to the opposite orientation. In this article, we study several spectral properties of distance Laplacian matrices of $\mathbb{T}$-gain graphs. In particular, we establish a characterization for the balanced $ \mathbb{T}$-gain graph in terms of the nullity of gain distance Laplacian matrices. A…

Continuous-Time Quantum Markov Chains And Discretizations Of p-Adic Schr\"odinger Equations: Comparisons And Simulations
W. A. Z\'u\~niga-Galindo, L. F. Chac\'on-Cort\'es
https://arxiv.org/abs/2508.06712

Continuous-Time Quantum Markov Chains And Discretizations Of p-Adic Schrödinger Equations: Comparisons And Simulations
The continuous-time quantum walks (CTQWs) are a fundamental tool in the development of quantum algorithms. Recently, it was shown that discretizations of p-adic Schrödinger equations give rise to continuous-time quantum Markov chains (CTQMCs); this type of Markov chain includes the CTQWs constructed using adjacency matrices of graphs as a particular case. In this paper, we study a large class of p-adic Schrödinger equations and the associated CTQMCs by comparing them with p-adic heat equation…

Constructing and Sampling Directed Graphs with Linearly Rescaled Degree Matrices
Yunxiang Yan, Meng Jiang
https://arxiv.org/abs/2507.23025 https://arxiv.or…

Constructing and Sampling Directed Graphs with Linearly Rescaled Degree Matrices
In recent years, many large directed networks such as online social networks are collected with the help of powerful data engineering and data storage techniques. Analyses of such networks attract significant attention from both the academics and industries. However, analyses of large directed networks are often time-consuming and expensive because the complexities of a lot of graph algorithms are often polynomial with the size of the graph. Hence, sampling algorithms that can generate graphs p…

The quantum Ramsey number $QR(2,2)$
Andrew Allen, Andre Kornell
https://arxiv.org/abs/2507.04128 https://arxiv.org/pdf/2507.04128

The quantum Ramsey number $QR(2,2)$
Operator systems of matrices can be viewed as quantum analogues of finite graphs. This analogy suggests many natural combinatorial questions in linear algebra. We determine the quantum Ramsey numbers $QR(2,k)$ and the lower quantum Turán numbers $T^\downarrow(n, m)$ with $n > 3m$. In particular, we conclude that $QR(2,2) = 4$ and confirm Weaver's conjecture that $T^\downarrow(4, 1) = 4$. We also obtain a new result for the existence of anticliques in quantum graphs of low dimension.

A unifying approach to self-organizing systems interacting via conservation laws
Frank Barrows, Guanming Zhang, Satyam Anand, Zizi Chen, Jonathan Lin, Amman Desai, Stefano Martiniani, Francesco Caravelli
https://arxiv.org/abs/2507.02575

A unifying approach to self-organizing systems interacting via conservation laws
We present a unified framework for embedding and analyzing dynamical systems using generalized projection operators rooted in local conservation laws. By representing physical, biological, and engineered systems as graphs with incidence and cycle matrices, we derive dual projection operators that decompose network fluxes and potentials. This formalism aligns with principles of non-equilibrium thermodynamics and captures a broad class of systems governed by flux-forcing relationships and local c…

Hook immanantal equalities for linear combination matrices of (di)graphs and their applications
Xiangshuai Dong, Tingzeng Wu, HongJian Lai
https://arxiv.org/abs/2508.17743 https…

Hook immanantal equalities for linear combination matrices of (di)graphs and their applications
Let $χ_λ$ be an irreducible character of the symmetric group $S_n$. For an $n \times n$ matrix $M = (m_{ij})$, define the immanant of $M$ corresponding to $χ_λ$ by \begin{eqnarray*} d_λ(M) = \sum_{σ\in S_n} χ_λ(σ) \prod_{i=1}^n m_{iσ(i)}. \end{eqnarray*} For $λ= (k, 1^{n-k})$, the immanant $d_{(k, 1^{n-k})}(M)$ is called the hook immanant and denoted by $d_k(M)$. The hook immanant polynomial of matrix $M$ is defined as $d_{k}(xI_n - M)$, where $I_n$ is the $n \times n$ identity matri…