Decomposition theorem on matchable distributive lattices

A distributive lattice structure ${\mathbf M}(G)$ has been established on the set of perfect matchings of a plane bipartite graph $G$. We call a lattice {\em matchable distributive lattice} (simply MDL) if it is isomorphic to such a distributive lattice. It is natural to ask which lattices are MDLs. We show that if a plane bipartite graph $G$ is elementary, then ${\mathbf M}(G)$ is irreducible. Based on this result, a decomposition theorem on MDLs is obtained: a finite distributive lattice $\mathbf{L}$ is an MDL if and only if each factor in any cartesian product decomposition of $\mathbf{L}$ is an MDL. Two types of MDLs are presented: $J(\mathbf{m}\times \mathbf{n})$ and $J(\mathbf{T})$, where $\mathbf{m}\times \mathbf{n}$ denotes the cartesian product between $m$-element chain and $n$-element chain, and $\mathbf{T}$ is a poset implied by any orientation of a tree.Comment: 19 pages, 7 figure

The Interaction Between PDE and Graphs in Multiscale Modeling

In this article an upscaled model is presented, for complex networks with highly clustered regions exchanging some abstract quantities in both, microscale and macroscale level. Such an intricate system is approximated by a partitioned open map in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$. The behavior of the quantities is modeled as flowing in the map constructed and thus it is subject to be described by partial differential equations. We follow this approach using the Darcy Porous Media, saturated fluid flow model in mixed variational formulation.Comment: 14 pages, 4 figure

Fullerenes with the maximum Clar number

The Clar number of a fullerene is the maximum number of independent resonant hexagons in the fullerene. It is known that the Clar number of a fullerene with n vertices is bounded above by [n/6]-2. We find that there are no fullerenes whose order n is congruent to 2 modulo 6 attaining this bound. In other words, the Clar number for a fullerene whose order n is congruent to 2 modulo 6 is bounded above by [n/6]-3. Moreover, we show that two experimentally produced fullerenes C80:1 (D5d) and C80:2 (D2) attain this bound. Finally, we present a graph-theoretical characterization for fullerenes, whose order n is congruent to 2 (respectively, 4) modulo 6, achieving the maximum Clar number [n/6]-3 (respectively, [n/6]-2)

Scattering Amplitudes and Toric Geometry

In this paper we provide a first attempt towards a toric geometric interpretation of scattering amplitudes. In recent investigations it has indeed been proposed that the all-loop integrand of planar N=4 SYM can be represented in terms of well defined finite objects called on-shell diagrams drawn on disks. Furthermore it has been shown that the physical information of on-shell diagrams is encoded in the geometry of auxiliary algebraic varieties called the totally non negative Grassmannians. In this new formulation the infinite dimensional symmetry of the theory is manifest and many results, that are quite tricky to obtain in terms of the standard Lagrangian formulation of the theory, are instead manifest. In this paper, elaborating on previous results, we provide another picture of the scattering amplitudes in terms of toric geometry. In particular we describe in detail the toric varieties associated to an on-shell diagram, how the singularities of the amplitudes are encoded in some subspaces of the toric variety, and how this picture maps onto the Grassmannian description. Eventually we discuss the action of cluster transformations on the toric varieties. The hope is to provide an alternative description of the scattering amplitudes that could contribute in the developing of this very interesting field of research.Comment: 58 pages, 25 figures, typos corrected, a reference added, to be published in JHE

Resonance graphs of plane bipartite graphs as daisy cubes

We characterize all plane bipartite graphs whose resonance graphs are daisy cubes and therefore generalize related results on resonance graphs of benzenoid graphs, catacondensed even ring systems, as well as 2-connected outerplane bipartite graphs. Firstly, we prove that if $G$ is a plane elementary bipartite graph other than $K_2$, then the resonance graph $R(G)$ is a daisy cube if and only if the Fries number of $G$ equals the number of finite faces of $G$, which in turn is equivalent to $G$ being homeomorphically peripheral color alternating. Next, we extend the above characterization from plane elementary bipartite graphs to all plane bipartite graphs and show that the resonance graph of a plane bipartite graph $G$ is a daisy cube if and only if $G$ is weakly elementary bipartite and every elementary component of $G$ other than $K_2$ is homeomorphically peripheral color alternating. Along the way, we prove that a Cartesian product graph is a daisy cube if and only if all of its nontrivial factors are daisy cubes