On the Representability of Line Graphs

A graph G=(V,E) is representable if there exists a word W over the alphabet V such that letters x and y alternate in W if and only if (x,y) is in E for each x not equal to y. The motivation to study representable graphs came from algebra, but this subject is interesting from graph theoretical, computer science, and combinatorics on words points of view. In this paper, we prove that for n greater than 3, the line graph of an n-wheel is non-representable. This not only provides a new construction of non-representable graphs, but also answers an open question on representability of the line graph of the 5-wheel, the minimal non-representable graph. Moreover, we show that for n greater than 4, the line graph of the complete graph is also non-representable. We then use these facts to prove that given a graph G which is not a cycle, a path or a claw graph, the graph obtained by taking the line graph of G k-times is guaranteed to be non-representable for k greater than 3.Comment: 10 pages, 5 figure

Restricted non-separable planar maps and some pattern avoiding permutations

Tutte founded the theory of enumeration of planar maps in a series of papers in the 1960s. Rooted non-separable planar maps are in bijection with West-2-stack-sortable permutations, beta(1,0)-trees introduced by Cori, Jacquard and Schaeffer in 1997, as well as a family of permutations defined by the avoidance of two four letter patterns. In this paper we give upper and lower bounds on the number of multiple-edge-free rooted non-separable planar maps. We also use the bijection between rooted non-separable planar maps and a certain class of permutations, found by Claesson, Kitaev and Steingrimsson in 2009, to show that the number of 2-faces (excluding the root-face) in a map equals the number of occurrences of a certain mesh pattern in the permutations. We further show that this number is also the number of nodes in the corresponding beta(1,0)-tree that are single children with maximum label. Finally, we give asymptotics for some of our enumerative results.Comment: 18 pages, 14 figure

The Discrete Fundamental Group of the Associahedron, and the Exchange Module

The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory. We study the abelianization of the discrete fundamental group, and show that it is free abelian of rank $\binom{n+2}{4}$. We also find a combinatorial description for a basis of this rank. We also introduce the exchange module of the type $A_n$ cluster algebra, used to model the relations in the cluster algebra. We use the discrete fundamental group to the study of exchange module, and show that it is also free abelian of rank $\binom{n+2}{3}$.Comment: 16 pages, 4 figure

The Discrete Fundamental Group of the Associahedron

The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory, that is we consider 5-cycles in the 1-skeleton of the associahedron to be combinatorial holes, but 4-cycles to be contractible. We give a simple description of the equivalence classes of 5-cycles in the 1-skeleton and then identify a set of 5-cycles from which we may produce all other cycles. This set of 5-cycle equivalence classes turns out to be the generating set for the abelianization of the discrete fundamental group of the associahedron. In this paper we provide presentations for the discrete fundamental group and the abelianization of the discrete fundamental group. We also discuss applications to cluster algebras as well as generalizations to type B and D associahedra. \pa

k-Parabolic Subspace Arrangements

In this paper, we study k-parabolic arrangements, a generalization of the k-equal arrangement for any finite real reflection group. When k=2, these arrangements correspond to the well-studied Coxeter arrangements. Brieskorn (1971) showed that the fundamental group of the complement of the type W Coxeter arrangement (over $\mathbb{C}$) is isomorphic to the pure Artin group of type W. Khovanov (1996) gave an algebraic description for the fundamental group of the complement of the 3-equal arrangement (over $\mathbb{R}$). We generalize Khovanov's result to obtain an algebraic description of the fundamental group of the complement of the 3-parabolic arrangement for arbitrary finite reflection group. Our description is a real analogue to Brieskorn's description

Synergy Between Intercellular Communication and Intracellular Ca2+ Handling in Arrhythmogenesis

Calcium is the primary signalling component of excitation-contraction coupling, the process linking electrical excitability of cardiac muscle cells to coordinated contraction of the heart. Understanding Ca2þ handling processes at the cellular level and the role of intercellular communication in the emergence of multicellular synchronization are key aspects in the study of arrhythmias. To probe these mechanisms, we have simulated cellular interactions on large scale arrays that mimic cardiac tissue, and where individual cells are represented by a mathematical model of intracellular Ca2þ dynamics. Theoretical predictions successfully reproduced experimental findings and provide novel insights on the action of two pharmacological agents (ionomycin and verapamil) that modulate Ca2þ signalling pathways via distinct mechanisms. Computational results have demonstrated how transitions between local synchronisation events and large scale wave formation are affected by these agents. Entrainment phenomena are shown to be linked to both ntracellular Ca2þ and coupling-specific dynamics in a synergistic manner. The intrinsic variability of the cellular matrix is also shown to affect emergent patterns of rhythmicity, providing insights into the origins of arrhythmogenic Ca2þ perturbations in cardiac tissue in situ

The Discrete Fundamental Group of the Associahedron

The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory, that is we consider 5-cycles in the 1-skeleton of the associahedron to be combinatorial holes, but 4-cycles to be contractible. We give a simple description of the equivalence classes of 5-cycles in the 1-skeleton and then identify a set of 5-cycles from which we may produce all other cycles. This set of 5-cycle equivalence classes turns out to be the generating set for the abelianization of the discrete fundamental group of the associahedron. In this paper we provide presentations for the discrete fundamental group and the abelianization of the discrete fundamental group. We also discuss applications to cluster algebras as well as generalizations to type B and D associahedra. \parL'associahèdre est un objet bien etudié que l'on retrouve dans plusieurs contextes. Par exemple, il est associé à la théorie des opérades, à l'étude des partitions non-croisées, à la théorie des treillis et plus récemment aux algèbres dámas. Nous étudions cet objet par le biais de la théorie des homotopies discretes. En bref cette théorie signifie qu'un cycle de longueur 5 (sur le squelette de l'associahèdre) est considéré comme étant le bord d'un trou combinatoire, alors qu'un cycle de longueur 4 peut être contracté sans problème. Les classes d'homotopies discrètes sont donc des classes d'équivalence de cycles de longueurs 5. Nous donnons une description simple de ces classes d'équivalence et identifions un ensemble de générateurs du groupe correspondant (abélien) d'homotopies discrètes. Nous d'ecrivons également les liens entre notre construction et les algèbres d'amas