scholarly journals Coherent fans in the space of flows in framed graphs

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Vladimir I. Danilov ◽  
Alexander V. Karzanov ◽  
Gleb A. Koshevoy
Keyword(s):  
Simplicial Complex ◽  
Binary Relation ◽  
Directed Graph ◽  
The Other ◽  
Cluster Algebras ◽  
Linear Orders ◽  
International Audience ◽  
Acyclic Directed Graph ◽  
Extreme Rays ◽  
Space Of Flows

International audience Let $G=(V,E)$ be a finite acyclic directed graph. Being motivated by a study of certain aspects of cluster algebras, we are interested in a class of triangulations of the cone of non-negative flows in $G, \mathcal F_+(G)$. To construct a triangulation, we fix a raming at each inner vertex $v$ of $G$, which consists of two linear orders: one on the set of incoming edges, and the other on the set of outgoing edges of $v$. A digraph $G$ endowed with a framing at each inner vertex is called $framed$. Given a framing on $G$, we define a reflexive and symmetric binary relation on the set of extreme rays of $\mathcal F_+ (G)$. We prove that that the complex of cliques formed by this binary relation is a pure simplicial complex, and that the cones spanned by cliques constitute a unimodular simplicial regular fan $Σ (G)$ covering the entire $\mathcal F_+(G)$. Soit $G=(V,E)$ un graphe orientè, fini et acyclique. Nous nous intéressons, en lien avec l’étude de certains aspects des algèbres amassées, à une classe de triangulations du cône des flots positifs de $G, \mathcal F_+(G)$. Pour construire une triangulation, nous ajoutons une structure en chaque sommet interne $v$ de $G$, constituée de deux ordres totaux : l'un sur l'ensemble des arcs entrants, l'autre sur l'ensemble des arcs sortants de $v$. On dit alors que $G$ est structurè. On définit ensuite une relation binaire réflexive et symétrique sur l'ensemble des rayons extrêmes de $\mathcal F_+ (G)$. Nous démontrons que le complexe des cliques formè par cette relation binaire est un complexe simplicial pur, et que le cône engendré par les cliques forme un éventail régulier simplicial unimodulaire $Σ (G)$ qui couvre complètement $\mathcal F_+(G)$.

scholarly journals Cyclic inclusion-exclusion and the kernel of P -partitions

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Valentin Féray
Keyword(s):  
Directed Graph ◽  
Symmetric Function ◽  
Bipartite Graphs ◽  
Linear Map ◽  
Generating Series ◽  
International Audience ◽  
Decreasing Functions ◽  
Acyclic Directed Graph

International audience Following the lead of Stanley and Gessel, we consider a linear map which associates to an acyclic directed graph (or a poset) a quasi-symmetric function. The latter is naturally defined as multivariate generating series of non-decreasing functions on the graph (or of P -partitions of the poset).We describe the kernel of this linear map, using a simple combinatorial operation that we call cyclic inclusion- exclusion. Our result also holds for the natural non-commutative analog and for the commutative and non-commutative restrictions to bipartite graphs.

scholarly journals Generalised cluster algebras and $q$-characters at roots of unity

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Anne-Sophie Gleitz
Keyword(s):  
Cluster Algebra ◽  
Integral Form ◽  
The Other ◽  
Cluster Algebras ◽  
Roots Of Unity ◽  
Quantum Affine Algebra ◽  
Affine Algebra ◽  
Grothendieck Ring ◽  
Root Of Unity ◽  
International Audience

International audience Shapiro and Chekhov (2011) have introduced the notion of <i>generalised cluster algebra</i>; we focus on an example in type $C_n$. On the other hand, Chari and Pressley (1997), as well as Frenkel and Mukhin (2002), have studied the <i>restricted integral form</i> $U^{\mathtt{res}}_ε (\widehat{\mathfrak{g}})$ of a quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ where $q=ε$ is a root of unity. Our main result states that the Grothendieck ring of a tensor subcategory $C_{ε^\mathbb{z}}$ of representations of $U^{\mathtt{res}}_ε (L\mathfrak{sl}_2)$ is a generalised cluster algebra of type $C_{l−1}$, where $l$ is the order of $ε^2$. We also state a conjecture for $U^{\mathtt{res}}_ε (L\mathfrak{sl}_3)$, and sketch a proof for $l=2$. Shapiro et Chekhov (2011) ont introduit la notion d'<i>algèbre amassée généralisée</i>; nous étudions un exemple en type $C_n$. Par ailleurs, Chari et Pressley (1997), ainsi que Frenkel et Mukhin (2002), ont étudié la <i>forme entière restreinte</i> $U^{\mathtt{res}}_ε (\widehat{\mathfrak{g}})$ d'une algèbre affine quantique $U_q(\widehat{\mathfrak{g}})$ où $q=ε$ est une racine de l'unité. Notre résultat principal affirme que l'anneau de Grothendieck d'une sous-catégorie tensorielle $C_{ε^\mathbb{z}}$ de représentations de $U^{\mathtt{res}}_ε (L\mathfrak{sl}_2)$ est une algèbre amassée généralisée de type $C_{l−1}$, où $l$ est l'ordre de $ε^2$. Nous conjecturons une propriété similaire pour $U^{\mathtt{res}}_ε (L\mathfrak{sl}_3)$ et donnons un aperçu de la preuve pour $l=2$.

scholarly journals Diameters and geodesic properties of generalizations of the associahedron

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
C. Ceballos ◽  
T. Manneville ◽  
V. Pilaud ◽  
L. Pournin
Keyword(s):  
The Other ◽  
Cluster Algebras ◽  
Other Hand ◽  
International Audience ◽  
The One ◽  
Generalized Associahedra

International audience The $n$-dimensional associahedron is a polytope whose vertices correspond to triangulations of a convex $(n + 3)$-gon and whose edges are flips between them. It was recently shown that the diameter of this polytope is $2n - 4$ as soon as $n > 9$. We study the diameters of the graphs of relevant generalizations of the associahedron: on the one hand the generalized associahedra arising from cluster algebras, and on the other hand the graph associahedra and nestohedra. Related to the diameter, we investigate the non-leaving-face property for these polytopes, which asserts that every geodesic connecting two vertices in the graph of the polytope stays in the minimal face containing both. L’associaèdre de dimension $n$ est un polytope dont les sommets correspondent aux triangulations d’un $(n + 3)$-gone convexe et dont les arêtes sont les échanges entre ces triangulations. Il a été récemment démontré que le diamètre de ce polytope est $2n - 4$ dès que $n > 9$. Nous étudions les diamètres des graphes de certaines généralisations de l’associaèdre : d’une part les associaèdres généralisés provenant des algèbres amassées, et d’autre part les associaèdres de graphes et les nestoèdres. En lien avec le diamètre, nous étudions si toutes les géodésiques entre deux sommets de ces polytopes restent dans la plus petite face les contenant.

scholarly journals Unification of Graphs and Relations in Mizar

2020 ◽  
Vol 28 (2) ◽  
pp. 173-186
Author(s):  
Sebastian Koch
Keyword(s):  
Binary Relation ◽  
The Other ◽  
Other Hand ◽  
System 2 ◽  
Definition Of

Summary A (di)graph without parallel edges can simply be represented by a binary relation of the vertices and on the other hand, any binary relation can be expressed as such a graph. In this article, this correspondence is formalized in the Mizar system [2], based on the formalization of graphs in [6] and relations in [11], [12]. Notably, a new definition of createGraph will be given, taking only a non empty set V and a binary relation E ⊆ V × V to create a (di)graph without parallel edges, which will provide to be very useful in future articles.

scholarly journals The Directed Forest Complex of Cayley Graphs

2020 ◽  
Author(s):  
Kennedy Courtney
Keyword(s):  
Simplicial Complex ◽  
Directed Graph ◽  
Finite Groups ◽  
Cayley Graphs

Let Γ be a directed graph. The directed forest complex, DF(Γ), is a simplicial complex whose vertices are the edges of Γ and whose simplices are sets of edges that form a directed forest in Γ. We study the directed forest complex of Cayley graphs of finite groups. The homology of DF(Γ) contains information about the graph, Γ and about the group, G. The ultimate goal is to classify DF(Γ) up to homotopy, compute its homology, and interpret the findings in terms of properties of DF(Γ). In this thesis, we present progress made toward this goal.

Complexity

1995 ◽  
Author(s):  
Raymond Greenlaw ◽  
H. James Hoover ◽  
Walter L. Ruzzo
Keyword(s):  
Directed Graph ◽  
Decision Problem ◽  
Flow Problem ◽  
Maximum Flow ◽  
The Other ◽  
Complexity Classes ◽  
Search Problem ◽  
Specific Question ◽  
Key Concepts ◽  
Formal Basis

The goal of this chapter is to provide the formal basis for many key concepts that are used throughout the book. These include the notions of problem, definitions of important complexity classes, reducibility, and completeness, among others. Thus far, we have used the term "problem" somewhat vaguely. In order to compare the difficulty of various problems we need to make this concept precise. Problems typically come in two flavors: search problems and decision problems. Consider the following search problem, to find the value of the maximum flow in a network. Example 3.1.1 Maximum Flow Value (MaxFlow-V) Given: A directed graph G = (V,E) with each edge e labeled by an integer capacity c(e) ≥ 0, and two distinguished vertices, s and t. Problem: Compute the value of the maximum flow from source s to sink t in G. The problem requires us to compute a number — the value of the maximum flow. Note, in this case we are actually computing a function. Now consider a variant of this problem. Example 3.1.2 Maximum Flow Bit (MaxFlow-B) Given: A directed graph G = (V, E) with each edge e labeled by an integer capacity c(e)≥ 0, and two distinguished vertices, s and t, and an integer i. Problem: Is the ith bit of the value of the maximum flow from source s to sink t in G a 1? This is a decision problem version of the flow problem. Rather than asking for the computation of some value, the problem is asking for a "yes" or "no" answer to a specific question. Yet the decision problem MaxFlow-B is equivalent to the search problem MaxFlow-V in the sense that if one can be solved efficiently in parallel, so can the other. Why is this? First consider how solving an instance of MaxFlow-B can be reduced to solving an instance of MaxFlow-V. Suppose that you are asked a question for MaxFlow-B, that is, "Is bit i of the maximum flow a 1?" It is easy to answer this question by solving MaxFlow-V and then looking at bit i of the flow.

scholarly journals The structure of intrinsic complexity of learning

1997 ◽  
Vol 62 (4) ◽  
pp. 1187-1201 ◽  
Author(s):  
Sanjay Jain ◽  
Arun Sharma
Keyword(s):  
Directed Graph ◽  
Learning Algorithm ◽  
Recursion Theory ◽  
Language Identification ◽  
Learning Problems ◽  
Pattern Languages ◽  
Language Classes ◽  
Resource Requirements ◽  
Intrinsic Complexity ◽  
Acyclic Directed Graph

AbstractLimiting identification of r.e. indexes for r.e. languages (from a presentation of elements of the language) and limiting identification of programs for computable functions (from a graph of the function) have served as models for investigating the boundaries of learnability. Recently, a new approach to the study of “intrinsic” complexity of identification in the limit has been proposed. This approach, instead of dealing with the resource requirements of the learning algorithm, uses the notion of reducibility from recursion theory to compare and to capture the intuitive difficulty of learning various classes of concepts. Freivalds, Kinber, and Smith have studied this approach for function identification and Jain and Sharma have studied it for language identification.The present paper explores the structure of these reducibilities in the context of language identification. It is shown that there is an infinite hierarchy of language classes that represent learning problems of increasing difficulty. It is also shown that the language classes in this hierarchy are incomparable, under the reductions introduced, to the collection of pattern languages.Richness of the structure of intrinsic complexity is demonstrated by proving that any finite, acyclic, directed graph can be embedded in the reducibility structure. However, it is also established that this structure is not dense. The question of embedding any infinite, acyclic, directed graph is open.

scholarly journals CLUSTER AUTOMORPHISMS AND COMPATIBILITY OF CLUSTER VARIABLES

2014 ◽  
Vol 56 (3) ◽  
pp. 705-720 ◽  
Author(s):  
IBRAHIM ASSEM ◽  
VASILISA SHRAMCHENKO ◽  
RALF SCHIFFLER
Keyword(s):  
Cluster Algebra ◽  
The Other ◽  
Cluster Algebras ◽  
Dynkin Type ◽  
Rank 2

AbstractIn this paper, we introduce a notion of unistructural cluster algebras, for which the set of cluster variables uniquely determines the clusters, as well as the notion of weak unistructural cluster algebras, for which the set of cluster variables determines the clusters, provided that the type of the cluster algebra is fixed. We prove that, for cluster algebras of the Dynkin type, the two notions of unistructural and weakly unistructural coincide, and that cluster algebras of rank 2 are always unistructural. We then prove that a cluster algebra $\mathcal A$ is weakly unistructural if and only if any automorphism of the ambient field, which restricts to a permutation of cluster variables of $\mathcal A$, is a cluster automorphism. We also investigate the Fomin-Zelevinsky conjecture that two cluster variables are compatible if and only if one does not appear in the denominator of the Laurent expansions of the other.

scholarly journals Cluster algebras of unpunctured surfaces and snake graphs

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Gregg Musiker ◽  
Ralf Schiffler
Keyword(s):  
Perfect Matchings ◽  
Laurent Polynomial ◽  
Cluster Algebras ◽  
Polynomial Expansion ◽  
International Audience ◽  
Polynomial Expansions

International audience We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph $G_{T,\gamma}$ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph $G_{T,\gamma}$ . Nous étudions des algèbres amassées avec coefficients principaux associées aux surfaces. Nous présentons une formule directe pour les développements de Laurent des variables amassées dans ces algèbres en terme de couplages parfaits d'un certain graphe $G_{T,\gamma}$ que l'on construit a partir de la surface en recollant des pièces élémentaires que l'on appelle carreaux. Nous donnons aussi une seconde formule pour ces développements en termes de sous-graphes de $G_{T,\gamma}$ .

Sign in / Sign up

Export Citation Format

Share Document