In this paper, we define a Hopf algebra structure on the vector space spanned by packed words using a selection-quotient coproduct. We show that this algebra is free on its irreducible packed words. Finally, we give some brief explanations on the Maple codes we have used.
In this paper, we define a Hopf algebra structure on the vector space spanned by packed words using a selection/quotient coproduct. We show that this algebra is free on its irreducible packed words. We also construct the Hilbert series of this Hopf algebra and we investigate its primitive elements.
We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra indu
ces a natural non-trivial sub-Hopf algebra on $c$-clusters in the theory of cluster algebras.
A non-commutative, planar, Hopf algebra of rooted trees was proposed in L. Foissy, Bull. Sci. Math. 126 (2002) 193-239. In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a qua
ntum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators).
We illustrate connections between differential geometry on finite simple graphs G=(V,E) and Riemannian manifolds (M,g). The link is that curvature can be defined integral geometrically as an expectation in a probability space of Poincare-Hopf indices
of coloring or Morse functions. Regge calculus with an isometric Nash embedding links then the Gauss-Bonnet-Chern integrand of a Riemannian manifold with the graph curvature. There is also a direct nonstandard approach: if V is a finite set containing all standard points of M and E contains pairs which are infinitesimally close in the sense of internal set theory, one gets a finite simple graph (V,E) which gets a curvature which as a measure corresponds to the standard curvature. The probabilistic approach is an umbrella framework which covers discrete spaces, piecewise linear spaces, manifolds or varieties.
Switches are operations which make local changes to the edges of a graph, usually with the aim of preserving the vertex degrees. We study a restricted set of switches, called triangle switches. Each triangle switch creates or deletes at least one tri
angle. Triangle switches can be used to define Markov chains which generate graphs with a given degree sequence and with many more triangles (3-cycles) than is typical in a uniformly random graph with the same degrees. We show that the set of triangle switches connects the set of all $d$-regular graphs on $n$ vertices, for all $dgeq 3$. Hence, any Markov chain which assigns positive probability to all triangle switches is irreducible on these graphs. We also investigate this question for 2-regular graphs.