Related papers: Integrable systems on quad-graphs
Singular complex analytic vector fields on the Riemann surfaces enjoy several geometric properties (singular means that poles and essential singularities are admissible). We describe relations between singular complex analytic vector fields…
In this dissertation, we explore the structure of inversion graphs of permutations--a class of graphs that naturally arises by representing each permutation as a graph, where vertices correspond to entries and edges encode inversions.…
In this survey, we explore recent literature on finding the cores of higher graphs using geometric and topological means. We study graphs, hypergraphs, and simplicial complexes, all of which are models of higher graphs. We study the notion…
Steinerberger proposed a notion of curvature on graphs involving the graph distance matrix (J. Graph Theory, 2023). We show that nonnegative curvature is almost preserved under three graph operations. We characterize the distance matrix and…
Scaled relative graphs have been originally introduced in the context of convex optimization and have recently gained attention in the control systems community for the graphical analysis of nonlinear systems. Of particular interest in…
This paper establishes three relations between the Toda field theory associated to a simple Lie algebra and the integral curves of the standard differential system on the corresponding complete flag variety. The motivation comes from the…
The group of automorphisms of the geometry of an integrable system is considered. The geometrical structure used to obtain it is provided by a normal form representation of integrable systems that do not depend on any additional geometrical…
We take an elementary and systematic approach to the problem of extending the Tutte polynomial to the setting of embedded graphs. Four notions of embedded graphs arise naturally when considering deletion and contraction operations on graphs…
We establish the equivalence of a 2D contour dynamics to the dispersionless limit of the integrable Toda hierarchy constrained by a string equation. Remarkably, the same hierarchy underlies 2D quantum gravity.
Graph theoretical ideas are highly utilized by computer science fields especially data mining. In this field, a data structure can be designed in the form of tree. Covering is a widely used form of data representation in data mining and…
The study of graph drawings on 2-surfaces is an active area of mathematical research. Our main results are criteria for integer and modulo 2 embeddability of graphs to surfaces.
We investigate how exchange-graph techniques can be used to study the topology of strata of meromorphic quadratic differentials. The exchange graph provides natural generators for the fundamental group. By extending the combinatorics of…
Graph-level representations are crucial tools for characterising structural differences between graphs. However, comparing graphs with different cardinalities, even when sampled from the same underlying distribution, remains challenging.…
There is a profound connection between copositive matrices and graph theory. Copositive matrices provide a powerful tool for formulating and solving various challenging graph-related problems. Conversely, graph theory provides a rich set of…
Decomposable dependency models and their graphical counterparts, i.e., chordal graphs, possess a number of interesting and useful properties. On the basis of two characterizations of decomposable models in terms of independence…
We present a method of constructing discrete integrable systems with crystallographic reflection group (Weyl) symmetries, thus clarifying the relationship between different discrete integrable systems in terms of their symmetry groups.…
This paper develops a symplectic bifurcation theory for integrable systems in dimension four. We prove that if an integrable system has no hyperbolic singularities and its bifurcation diagram has no vertical tangencies, then the fibers of…
We extend the notion of graph homomorphism to cellularly embedded graphs (maps) by designing operations on vertices and edges that respect the surface topology; we thus obtain the first definition of map homomorphism that preserves both the…
We describe optimal robust algorithms for finding a triangle and the unweighted girth in a unit disk graph, as well as finding a triangle in a transmission graph.In the robust setting, the input is not given as a set of sites in the plane,…
We study relations between geometric embeddings of graphs and the spectrum of associated matrices, focusing on outerplanar embeddings of graphs. For a simple connected graph $G=(V,E)$, we define a "good" $G$-matrix as a $V\times V$ matrix…