arXiv research

Extends graph factor system to quasi-median graphs.

The paper finds minimum Steklov eigenvalues on combinatorial graphs.

Combinatorial approach to compute satellite knot invariants using graph theory.

We introduce and study combinatorial equivariant analogues of the Kronheimer--Mrowka homology theory of planar trivalent graphs.

We propose a new family of combinatorial inference problems for graphical models. Unlike classical statistical inference where the main interest is point estimation or parameter testing, combinatorial inference aims at testing the global structure of the underlying graph. Examples include testing the graph connectivity…

BIG Laplacians bridge combinatorial and Hodge Laplacians for discrete data.

A planar graph is inscribable if it is combinatorial equivalent to the skeleton of a polyhedra which is inscribed in a sphere. For an inscribable graph, in its combinatorial equivalent class, if we could always find polyhedra inscribed in any given convex surface which is sufficiently close to the sphere, then we call …

We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks. The characterization is a true analogue of the Maxwell-Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorit…

New combinatorial structures represent subgroups of surface groups, analogous to Stallings core graphs.

This is a short review article on invariants of spatial graphs, written for "A Concise Encyclopedia of Knot Theory" (ed. Adams et. al.). The emphasis is on combinatorial and polynomial invariants of spatial graphs, including the Alexander polynomial, the fundamental quandle of a graph, and the Yamada polynomial.

The paper reveals a property of chromatic homology for complete graphs.

The present paper is an introduction to a combinatorial theory arising as a natural generalisation of classical and virtual knot theory. There is a way to encode links by a class of `realisable' graphs. When passing to generic graphs with the same equivalence relations we get `graph-links'. On one hand graph-links gene…

The paper compares Steklov and Laplacian eigenvalues on graphs.

The study shows how discrete graphs can resemble hypercube structures under certain curvature conditions.

In 2003, Ozsváth and Szabó defined the concordance invariant $τ$ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of $τ$ for knots in $S^3$ and a combinatorial proof that $τ$ gives a lower bound for the slice genus of a knot. Recently, Har…

In this paper, from a theoretical perspective, we study how powerful graph neural networks (GNNs) can be for learning approximation algorithms for combinatorial problems. To this end, we first establish a new class of GNNs that can solve a strictly wider variety of problems than existing GNNs. Then, we bridge the gap b…

Theory of symmetric rigidity in hyperbolic geometry.

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a compl…

The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general form of a Laplacian on …

End-to-end trainable graph matching using improved combinatorial solvers.

Extends graph similarity theory to improve MPNNs' generalization abilities.

This paper focuses on Bayesian Optimization (BO) for objectives on combinatorial search spaces, including ordinal and categorical variables. Despite the abundance of potential applications of Combinatorial BO, including chipset configuration search and neural architecture search, only a handful of methods have been pro…

Surveying machine learning for solving graph optimization problems.

Graph learning from data represents a canonical problem that has received substantial attention in the literature. However, insufficient work has been done in incorporating prior structural knowledge onto the learning of underlying graphical models from data. Learning a graph with a specific structure is essential for …

New combinatorial type helps distinguish plane curve topologies.

Combinatorial approach to $α$-Ricci and Lin-Lu-Yau Ricci curvatures on graphs

Graph neural networks improve combinatorial optimization by leveraging inductive bias.

Consider a finite, regular cover $Y\to X$ of finite graphs, with associated deck group $G$. We relate the topology of the cover to the structure of $H_1(Y;\mathbb{C})$ as a $G$-representation. A central object in this study is the {\em primitive homology} group $H_1^{\mathrm{prim}}(Y;\mathbb{C})\subseteq H_1(Y;\mathbb{…

Graph machine learning lacks a balanced theory, focusing on expressive power and optimization.

We present a simple combinatorial model for quasipositive surfaces and positive braids, based on embedded bipartite graphs. As a first application, we extend the well-known duality on standard diagrams of torus links to twisted torus links. We then introduce a combinatorial notion of adjacency for bipartite graph links…

Classical knot theory can be generalized to virtual knot theory and spatial graph theory. In 2007, Fleming and Mellor combined virtual knot theory and spatial graph theory to form, combinatorially, virtual spatial graph theory. In this paper, we introduce a topological definition of virtual spatial graphs that is simil…

Advances combinatorial complexes for better modeling of hierarchical and set-type relations.

Extends knot concordance invariant to balanced spatial graphs using grid homology.

We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of minimally rigid periodic frameworks with fixed-area fundamental domain and fixed-angle fundamental domain.

We prove several combinatorial results on path algebras over discrete structures related to directed graphs. These results are motivated by Morse theory on a manifold with boundary and, more generally, by Floer theory on a configuration space with boundary. Their purpose is to organize cobordism relationships among mod…

Graph coloring is explained using a topological field theory with defects.

Computes Khovanov homology for 2-strand braids via graph relations.

This work proposes an unsupervised neural network framework for solving combinatorial optimization problems on graphs.

Bayesian Optimization for graph node subset functions.

Linear-time graph optimization using reinforcement learning.

New method calculates discrete curvature using effective resistances.

The paper extends log-Sobolev inequalities to matrix-valued settings using combinatorial methods.

We prove that the total curvature of any planar graph with nonnegative combinatorial curvature is an integral multiple of $\frac{1}{12}.$ As a corollary, this answers a question proposed by T. Réti.

This paper extends combinatorial semi-bandits to graph feedback, improving regret bounds.

Optimizes matching in weighted graphs with semi-bandit sampling.

Graphs from van der Corput sequence embed into Chamanara surface.

A new deep learning framework for topological data.

Novel theory combines combinatorial and topological elements.