Related papers: Integrable systems on quad-graphs

This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose a curvature on the nodes and links of a graph and present the evidence for their interpretation as a curvature. Notably,…

In this paper, we analyze embeddings of grid graphs on orientable surfaces. We determine the genus of a large class of k-dimensional grid graphs and effective two-sided bounds for the genus of any 3-dimensional grid graph, both in terms of…

The objective of this work is to examine the integrability of Hamiltonian systems in $2D$ spaces with variable curvature of certain types. Based on the differential Galois theory, we announce the necessary conditions of the integrability.…

This monograph, written for educational purposes, serves as an introduction to the concept of integrability as it applies to systems of differential equations (both ordinary and partial) as well as to vector-valued fields. The general cases…

In this paper, we introduce a graph structure, called non-zero component graph on finite dimensional vector spaces. We show that the graph is connected and find its domination number and independence number. We also study the…

The Lie algebraic integrability test is applied to the problem of classification of integrable Klein-Gordon type equations on quad-graphs. The list of equations passing the test is presented containing several well-known integrable models.…

In this paper, we introduce a family of graphs which is a generalization of zero-divisor graphs and compute an upper-bound for the diameter of such graphs. We also investigate their cycles and cores.

Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…

How does one generalize differential geometric constructs such as curvature of a manifold to the discrete world of graphs and other combinatorial structures? This problem carries significant importance for analyzing models of discrete…

The knowledge of {\it non usual} and sometimes {\it hidden} symmetries of (classical) integrable systems provides a very powerful setting-out of solutions of these models. Primarily, the understanding and possibly the quantisation of…

Gauge invariance in discrete dynamical systems and its connection with quantization are considered. For a complete description of gauge symmetries of a system we construct explicitly a class of groups unifying in a natural way the space and…

We construct zero-curvature representations for the equations of motion of a class of sigma-models with complex homogeneous target spaces, not necessarily symmetric. We show that in the symmetric case the proposed flat connection is…

We give inductive constructions of independent graphs that contain implied nonedges but do not contain any non-trivial rigid subgraphs, or \emph{nucleations}: some of the constructions and proofs apply to 3-dimensional abstract rigidity…

Based on our previous graph covering method, we introduce weighted graph covering models and flexible graph covering models that are almost equivalent to the well-known Bratteli--Vershik models. These models play important roles in showing…

The periodic discrete Toda equation defined over finite fields has been studied. We obtained the finite graph structures constructed by the network of states where edges denote possible time evolutions. We simplify the graphs by introducing…

Using an exact holographic duality formula between the inhomogeneous 2d Ising model and 3d quantum gravity, we provide a formula for "real" zeroes of the 2d Ising partition function on finite trivalent graphs in terms of the geometry of a…

We determine the optimum topology of quasi-one dimensional nonlinear optical structures using generalized quantum graph models. Quantum graphs are relational graphs endowed with a metric and a multiparticle Hamiltonian acting on the edges,…

A list of forty third-order exactly integrable two-field evolutionary systems is presented. Differential substitutions connecting various systems from the list are found. It is proved that all the systems can be obtained from only two of…

Simplicial surfaces describe the incidence relations between vertices, edges and faces of triangulated 2-dimensional manifolds in a purely combinatorial way. By considering only the incidences of edges and faces, simplicial surfaces are…

This paper proposes a method for identifying and classifying integrable nonlinear equations with three independent variables, one of which is discrete and the other two are continuous. A characteristic property of this class of equations,…