Related papers: Geometric approach to graph magnitude homology

This paper presents several algorithms for hashing directed graphs. The algorithms given are capable of hashing entire graphs as well as assigning hash values to specific nodes in a given graph. The notion of node symmetry is made precise…

We consider symmetric powers of a graph. In particular, we show that the spectra of the symmetric square of strongly regular graphs with the same parameters are equal. We also provide some bounds on the spectra of the symmetric squares of…

We investigate associativity of multiplications on chain complexes over commutative noetherian rings from two perspectives. First, we introduce a natural associator subcomplex and show how its homology can detect associativity. Second, we…

For simple graphs $G$ and $H$, the Hom complex $\mathrm{Hom}(G,H)$ is a polyhedral complex whose vertices are the graph homomorphisms $G\to H$ and whose edges connect the pairs of homomorphisms which differ in a single vertex of $G$. Hom…

Let the join of two graphs be the union of two disjoint graphs connected by $j$ edges in a one-to-one manner. In previous work by Gyurov and Pinzon, which generalized the results of Badura and Rara, the determinant of the adjacency matrix…

We generalize the fundamental graph-theoretic notion of chordality for higher dimensional simplicial complexes by putting it into a proper context within homology theory. We generalize some of the classical results of graph chordality to…

A mixed graph can be seen as a type of digraph containing some edges (two opposite arcs). Here we introduce the concept of sequence mixed graphs, which is a generalization of both sequence graphs and iterated line digraphs. These structures…

We define for every dendroidal set X a chain complex and show that this assignment determines a left Quillen functor. Then we define the homology groups $H_n(X)$ as the homology groups of this chain complex. This generalizes the homology of…

An explicit isomorphism between Morse homology and singular homology is constructed via the technique of pseudo-cycles. Given a Morse cycle as a formal sum of critical points of a Morse function, the unstable manifolds for the negative…

Persistent homology is a method for computing the topological features present in a given data. Recently, there has been much interest in the integration of persistent homology as a computational step in neural networks or deep learning. In…

We compute the factorization homology of a polynomial algebra over a compact and closed manifold with trivialized tangent bundle up to weak equivalence in a new way. This calculation is based on the model of a graph complex and an explicit…

Magnitude homology is an invariant of enriched categories which generalizes ordinary categorical homology -- the homology of the classifying space of a small category. The classifying space can also be generalized in a different direction:…

Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging…

We give an overview of different approaches to measuring the similarity of, or the distance between, two graphs, highlighting connections between these approaches. We also discuss the complexity of computing the distances.

To a rational homology sphere graph manifold one can associate a weighted tree invariant called splice diagram. In this article we prove a sufficient numerical condition on the splice diagram for a graph manifold to be a singularity link.…

We consider two notions describing how one finite graph may be larger than another. Using them, we prove several theorems for such pairs that compare the number of spanning trees, the return probabilities of random walks, and the number of…

In this article, we analyze the structure and relationships between magnitude homology and Eulerian magnitude homology of finite graphs. Building on the work of Kaneta and Yoshinaga, Sazdanovic and Summers, and Asao and Izumihara, we…

In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph…

We introduce graphcodes, a novel multi-scale summary of the topological properties of a dataset that is based on the well-established theory of persistent homology. Graphcodes handle datasets that are filtered along two real-valued scale…

In this expository paper we present some ideas of algebraic topology (more precisely, of homology theory) in a language accessible to non-specialists in the area. A $1$-cycle in a graph is a set $C$ of edges such that every vertex is…