Related papers: Comments on Exchange Graphs in Cluster Algebras
This paper classifies the derivations of group algebras in terms of the generators and defining relations of the group. If $RG$ is a group ring, where $R$ is commutative and $S$ is a set of generators of $G$ then necessary and sufficient…
We study the problem of clustering nodes in a dynamic graph, where the connections between nodes and nodes' cluster memberships may change over time, e.g., due to community migration. We first propose a dynamic stochastic block model that…
We demonstrate that when a graph exhibits a specific type of symmetry, it satisfies the Union Closed Conjecture(UCC). Additionally, we show that certain graph classes, such as Cylindrical Grid Graphs and Torus Grid Graphs also satisfy the…
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large…
We develop a new class of clockwork theories with an augmented structure of the near-neighbour interactions along a one-dimensional closed chain. Such a topology leads to new and attractive features in addition to generating light states…
Endomorphisms of Weyl algebras are studied using bimodules. Initially, for a Weyl algebra over a field of characteristic zero, Bernstein's inequality implies that holonomic bimodules finitely generated from the right or left form a monoidal…
Considered as commutative algebras, cluster algebras can be very unpleasant objects. However, the first author introduced a condition known as "local acyclicity" which implies that cluster algebras behave reasonably. One of the earliest and…
A finite transitive permutation group is elusive if it contains no derangements of prime order. These groups are closely related to a longstanding open problem in algebraic graph theory known as the Polycirculant Conjecture, which asserts…
Modern graph or network datasets often contain rich structure that goes beyond simple pairwise connections between nodes. This calls for complex representations that can capture, for instance, edges of different types as well as so-called…
How do vertices exert influence in graph data? We develop a framework for edge clustering, a new method for exploratory data analysis that reveals how both vertices and edges collaboratively accomplish directed influence in graphs,…
We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system.
We focus on the algorithm underlying the main result of [A. Mestre, R. Oeckl, Generating loop graphs via Hopf algebra in quantum field theory. J. Math. Phys., 47, 122302, 2006]. This is an algebraic formula to generate all connected graphs…
We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space $V$, and with…
We develop a general theory of cluster categories, applying to a 2-Calabi-Yau extriangulated category $\mathcal{C}$ and cluster-tilting subcategory $\mathcal{T}$ satisfying only mild finiteness conditions. We show that the structure theory…
Transit functions serve not only as abstractions of betweenness and convexity but are also closely connected with clustering systems. Here, we investigate the canonical transit functions of binary clustering systems inspired by pyramids,…
Weakly separated collections arise in the cluster algebra derived from the Pl\"ucker coordinates on the nonnegative Grassmannian. Oh, Postnikov, and Speyer studied weakly separated collections over a general Grassmann necklace $\mathcal{I}$…
The complexity of deciding whether a clustered graph admits a clustered planar drawing is a long-standing open problem in the graph drawing research area. Several research efforts focus on a restricted version of this problem where the…
A known failing of many popular random graph models is that the Aldous-Hoover Theorem guarantees these graphs are dense with probability one; that is, the number of edges grows quadratically with the number of nodes. This behavior is…
We study the duality between the mutations and the initial-seed mutations in cluster algebras, where the initial-seed mutations are the transformations of rational expressions of cluster variables in terms of the initial cluster under the…
Graph clustering is a fundamental technique in data analysis with applications in many different fields. While there is a large body of work on clustering undirected graphs, the problem of clustering directed graphs is much less understood.…