相关论文: Algebraic Properties of Gaussian HTC-identifiable …
Linear structural equation models relate the components of a random vector using linear interdependencies and Gaussian noise. Each such model can be naturally associated with a mixed graph whose vertices correspond to the components of the…
An invariant for cospectral graphs is a property shared by all cospectral graphs. In this paper, we establish three novel arithmetic invariants for cospectral graphs, revealing deep connections between spectral properties and combinatorial…
A mixed graph $G$ is a graph obtained from a simple undirected graph by orientating a subset of edges. $G$ is self-converse if it is isomorphic to the graph obtained from $G$ by reversing each directed edge. For two mixed graphs $G$ and $H$…
A linear structural equation model relates random variables of interest and corresponding Gaussian noise terms via a linear equation system. Each such model can be represented by a mixed graph in which directed edges encode the linear…
The interaction between combinatorics and algebraic and differential geometry is very strong. While researching a problem of Hessian topology, we came across a series of identities of binomial coefficients, which are useful for proving a…
The deep interconnection between linear algebra and graph theory allows one to interpret classical matrix invariants through combinatorial structures. To each square matrix A over a commutative ring K, one can associate a weighted directed…
We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature…
In combinatorial commutative algebra and algebraic statistics many toric ideals are constructed from graphs. Keeping the categorical structure of graphs in mind we give previous results a more functorial context and generalize them by…
Associated to any hypergraph is a toric ideal encoding the algebraic relations among its edges. We study these ideals and the combinatorics of their minimal generators, and derive general degree bounds for both uniform and non-uniform…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
The observational characteristics of a linear structural equation model can be effectively described by polynomial constraints on the observed covariance matrix. However, these polynomials can be exponentially large, making them impractical…
Many machine learning algorithms for tabular data produce black-box models, which prevent users from understanding the rationale behind the model predictions. In their unconstrained form, graph neural networks fall into this category, and…
We consider structural equation models (SEMs), in which every variable is a function of a subset of the other variables and a stochastic error. Each such SEM is naturally associated with a directed graph describing the relationships between…
A graph $G$ is said to be chordal if it has no induced cycles of length four or more. In a recent preprint Culbertson, Guralnik, and Stiller give a new characterization of chordal graphs in terms of sequences of what they call…
The present work is concerned with characterizing some algebraic invariants of edge ideals of hypergraphs. To this aim, firstly, we introduce some kinds of combinatorial invariants similar to matching numbers for hypergraphs. Then we…
We use the characteristic polynomial of the Coxeter matrix of an algebra to complete the combinatorial classification of piecewise hereditary algebras which Happel gave in terms of the trace of the Coxeter matrix. We also give a…
If all the eigenvalues of the Hermitian-adjacency matrix of a mixed graph are integers, then the mixed graph is called \emph{H-integral}. If all the eigenvalues of the (0,1)-adjacency matrix of a mixed graph are \emph{Gaussian integers},…
Given a finite, simple, vertex-weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to…
As a higher analogue of the edge ideal of a graph, we study the $t$-connected ideal $\operatorname{J}_{t}$. This is the monomial ideal generated by the connected subsets of size $t$. For chordal graphs, we show that $\operatorname{J}_{t}$…
Two directed graphs are called covariance equivalent if they induce the same set of covariance matrices, up to a Lebesgue measure zero set, on the random variables of their associated linear structural equation models. For acyclic graphs,…