Related papers: Comments on Exchange Graphs in Cluster Algebras

We say that a digraph is essentially cyclic if its Laplacian spectrum is not completely real. The essential cyclicity implies the presence of directed cycles, but not vice versa. The problem of characterizing essential cyclicity in terms of…

Interaction graphs, such as those recording emails between individuals or transactions between institutions, tend to be sparse yet structured, and often grow in an unbounded manner. Such behavior can be well-captured by structured,…

We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, for example, principal coefficients.

The Union Closed Sets Conjecture is one of the most renowned problems in combinatorics. Its appeal lies in the simplicity of its statement contrasted with the potential complexity of its resolution. The conjecture posits that, in any union…

In this paper, we prove some combinatorial results on generalized cluster algebras. To be more precisely, we prove that (i) the seeds of a generalized cluster algebra $\mathcal A(\mathcal S)$ whose clusters contain particular cluster…

We study the root of unity degeneration of cluster algebras and quantum dilogarithm identities. We prove identities for the cyclic dilogarithm associated with a mutation sequence of a quiver, and as a consequence new identities for the…

Several cryptographic protocols constructed based on less-known algorithmic problems, such as those in non-commutative groups, group rings, semigroups, etc., which claim quantum security, have been broken through classical reduction methods…

Let $\mathcal{A}_{q}$ be an arbitrary quantum cluster algebra with principal coefficients. We give the fundamental relations between the quantum cluster variables arising from one-step mutations from the initial cluster in…

In this article, we introduce the notion of cluster automorphism of a given cluster algebra as a $\ZZ$-automorphism of the cluster algebra that sends a cluster to another and commutes with mutations. We study the group of cluster…

Clustering a graph, i.e., assigning its nodes to groups, is an important operation whose best known application is the discovery of communities in social networks. Graph clustering and community detection have traditionally focused on…

In this paper we study the algebra of graph invariants, focusing mainly on the invariants of simple graphs. All other invariants, such as sorted eigenvalues, degree sequences and canonical permutations, belong to this algebra. In fact,…

Clustering is a fundamental problem in network analysis that finds closely connected groups of nodes and separates them from other nodes in the graph, while link prediction is to predict whether two nodes in a network are likely to have a…

We offer a solution to a long-standing problem in the physics of networks, the creation of a plausible, solvable model of a network that displays clustering or transitivity -- the propensity for two neighbors of a network node also to be…

Graph clustering is a fundamental problem in unsupervised learning, with numerous applications in computer science and in analysing real-world data. In many real-world applications, we find that the clusters have a significant high-level…

We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in…

Originally, tangles were invented as an abstract tool in mathematical graph theory to prove the famous graph minor theorem. In this paper, we showcase the practical potential of tangles in machine learning applications. Given a collection…

Graph clustering is a fundamental computational problem with a number of applications in algorithm design, machine learning, data mining, and analysis of social networks. Over the past decades, researchers have proposed a number of…

Clustering is the propensity of nodes that share a common neighbour to be connected. It is ubiquitous in many networks but poses many modelling challenges. Clustering typically manifests itself by a higher than expected frequency of…

We consider the problem of clustering grouped data with possibly non-exchangeable groups whose dependencies can be characterized by a known directed acyclic graph. To allow the sharing of clusters among the non-exchangeable groups, we…

We demonstrate that a large class of one-dimensional quantum and classical exchange models can be described by the same type of graphs, namely Cayley graphs of the permutation group. Their well-studied spectral properties allow us to derive…