Related papers: Persistent commutative algebra on graphs and hyper…
Several algebraic properties of a binomial edge ideal $J_G$ can be interpreted in terms of combinatorial properties of its associated graph $G$. In particular, the so-called cut sets of a graph $G$, special sets of vertices that disconnect…
We introduce and study the concept which we call the splitting of a graph and compare algebraic properties of the edge ideals of graphs and those of their splitting graphs.
In recent years, cosmic shear has emerged as a powerful tool to study the statistical distribution of matter in our Universe. Apart from the standard two-point correlation functions, several alternative methods like peak count statistics…
In this paper, we introduce the concept of complementary edge ideals of graphs and study their algebraic properties and invariants.
Let $R$ be a finite commutative ring with identity, and let $P$ be a proper prime ideal of $R$. The prime ideal graph $\Gamma_P(R)$ has vertex set of $R\setminus\{0\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if…
Let $G$ be a simple graph on $n$ vertices, and let $J_G$ denotes the corresponding binomial edge ideal in $S=\mathbb{K}[x_1,\ldots,x_n,y_1,\ldots,y_n]$, where $\mathbb{K}$ is a field. We show that if a vertex satisfies a certain degree…
Adjacency polytopes, a.k.a. symmetric edge polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, and dynamical systems. In particular,…
Many attempts have been made in recent decades to integrate machine learning (ML) and topological data analysis. A prominent problem in applying persistent homology to ML tasks is finding a vector representation of a persistence diagram…
In this work, we present a generalization of extended persistent homology to filtrations of graded sub-groups by defining relative homology in this setting. Our work provides a more comprehensive and flexible approach to get an algebraic…
We focus in this paper on edge ideals associated to bipartite graphs and give a combinatorial characterization of those having regularity 3. When the regularity is strictly bigger than 3, we determine the first step $i$ in the minimal…
In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals…
Strongly stable monomial ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers…
Representational limits of message-passing graph neural networks (MP-GNNs), e.g., in terms of the Weisfeiler-Leman (WL) test for isomorphism, are well understood. Augmenting these graph models with topological features via persistent…
We study the Betti numbers of binomial edge ideal associated to some classes of graphs with large Castelnuovo-Mumford regularity. As an application we give several lower bounds of the Castelnuovo-Mumford regularity of arbitrary graphs…
Artificial neural networks can learn complex, salient data features to achieve a given task. On the opposite end of the spectrum, mathematically grounded methods such as topological data analysis allow users to design analysis pipelines…
We study the dependence of graded Betti numbers of monomial ideals on the characteristic of the base field. The examples we describe include bipartite ideals, Stanley--Reisner ideals of vertex-decomposable complexes and ideals with…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally, persistence is a central tool in topological data analysis, which examines…
We introduce persistence with an emphasis on its algebraic foundations, using the representation theory of posets. Linear representations of posets arise in several areas of mathematics, including the representation theory of quivers and…
We study the problem of deterministic approximate counting of matchings and independent sets in graphs of bounded connective constant. More generally, we consider the problem of evaluating the partition functions of the monomer-dimer model…