相关论文: Geometric approach to graph magnitude homology
We compute magnitude homology of various graphs using algebraic Morse theory. Specifically, we (1) give an alternative proof that trees are diagonal, (2) identify a new class of diagonal graphs, (3) prove that the icosahedral graph is…
The magnitude homology, introduced by R. Hepworth and S. Willerton, offers a topological invariant that enables the study of graph properties. Hypergraphs, being a generalization of graphs, serve as popular mathematical models for data with…
This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. Magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology,…
Hypergraph is a topological model for networks. In order to study the topology of hypergraphs, the homology of the associated simplicial complexes and the embedded homology have been invented. In this paper, we give some algorithms to…
We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes, and exemplify their…
In this article, we show that magnitude homology and path homology are closely related, and we give some applications. We define differentials ${\mathrm MH}^{\ell}_k(G) \longrightarrow {\mathrm MH}^{\ell-1}_{k-1}(G)$ between magnitude…
The magnitude of a graph can be thought of as an integer power series associated to a graph; Leinster introduced it using his idea of magnitude of a metric space. Here we introduce a bigraded homology theory for graphs which has the…
In this paper we introduce a path complex that can be regarded as a generalization of the notion of a simplicial complex. The main motivation for considering path complexes comes from directed graphs(digraphs). We obtain a new notion of the…
We compute numerically the homology of several graph complexes in low loop orders, extending previous results.
We define an algebraic setup of homology for hypergraphs, which defaults to simplicial homology in the case of graphs, and study its basic properties. As part of our study we define algebraic spanning trees of hypergraphs, along with…
In this paper tackle the problem of computing the ranks of certain eulerian magnitude homology groups of a graph G. First, we analyze the computational cost of our problem and prove that it is #W[1]-complete. Then we develop the first…
In this paper we explore the algebraic structure and combinatorial properties of eulerian magnitude homology. First, we analyze the diagonality conditions of eulerian magnitude homology, providing a characterization of complete graphs.…
This paper studies the magnitude homology groups of geodesic metric spaces. We start with a description of the second magnitude homology of a general metric space in terms of the zeroth homology groups of certain simplicial complexes. Then,…
Asao-Ivanov showed that magnitude homology is a Tor functor, hence we can compute it by giving a projective resolution of a certain module. In this article, we compute magnitude homology by constructing a minimal projective resolution. As a…
In this paper, we review a method for computing and parameterizing the set of homotopy classes of chain maps between two chain complexes. This is then applied to finding topologically meaningful maps between simplicial complexes, which in…
Hypergraphs have seen widespread application in network and data science communities in recent years. We present a survey of recent work to construct auxiliary structures from hypergraphs -- specifically simplicial, relative, and chain…
A matching complex of a simple graph $G$ is a simplicial complex with faces given by the matchings of $G$. The topology of matching complexes is mysterious; there are few graphs for which the homotopy type is known. Marietti and Testa…
A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. By [11], a hypergraph gives an associated simplicial complex. By [4], the embedded homology of a hypergraph is the homology of the infimum chain…
Magnitude is a numerical invariant of metric spaces and graphs, analogous, in a precise sense, to Euler characteristic. Magnitude homology is an algebraic invariant constructed to categorify magnitude. Among the important features of the…
We study the homology of simplicial and cubical sets with symmetries. These are simplicial and cubical sets with additional maps expressing the symmetries of simplices and cubes. We consider the chain complex computing the homology groups…