Related papers: Weight systems and invariants of graphs and embedd…
We construct a graph G such that any embedding of G into R^{3} contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H…
In this paper we completely characterize the graphs which have an edge weighted adjacency matrix belonging to the class of $n \times n$ involutions with spectrum equal to $\{ \lambda_1^{n-2}, \lambda_2^{2} \}$ for some $\lambda_1$ and some…
A communication network can be modeled as a directed connected graph with edge weights that characterize performance metrics such as loss and delay. Network tomography aims to infer these edge weights from their pathwise versions measured…
Graph representation learning (also called graph embeddings) is a popular technique for incorporating network structure into machine learning models. Unsupervised graph embedding methods aim to capture graph structure by learning a…
We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual knot invariants developed in arXiv:1509.00578. In particular, following the theory of quantum invariants we work with 'rotational' virtual…
A way to associate unweighted graphs from weighted ones is presented, such that linear stable equilibria of the Kuramoto homogeneous model associated to both graphs coincide, i.e., equilibria of the system $\dot\theta_i = \sum_{j \sim i}…
This paper treats some basic points in general relativity and in its perturbative analysis. Firstly a systematic classification of global SO(n) invariants, which appear in the weak-field expansion of n-dimensional gravitational theories, is…
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…
Many recent results in combinatorics concern the relationship between the size of a set and the number of distances determined by pairs of points in the set. One extension of this question considers configurations within the set with a…
Topology identification and inference of processes evolving over graphs arise in timely applications involving brain, transportation, financial, power, as well as social and information networks. This chapter provides an overview of graph…
Vector representations of graphs and relational structures, whether hand-crafted feature vectors or learned representations, enable us to apply standard data analysis and machine learning techniques to the structures. A wide range of…
We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of…
Most real-world networks are weighted graphs with the weight of the edges reflecting the relative importance of the connections. In this work, we study non degree dependent correlations between edge weights, generalizing thus the…
Graph transformations definable in logic can be described using the notion of transductions. By understanding transductions as a basic embedding mechanism, which captures the possibility of encoding one graph in another graph by means of…
We propose a gauge model of quantum electrodynamics (QED) and its nonabelian generalization from which we derive knot invariants such as the Jones polynomial. Our approach is inspired by the work of Witten who derived knot invariants from…
The correspondence between weighted undirected graphs and reversible Markov chains via vertex random walks is simple and well known. Leveraging this correspondence and ideas from the theory of dynamical systems, we study the structural…
Hypergraph is a powerful representation in several computer vision, machine learning and pattern recognition problems. In the last decade, many researchers have been keen to develop different hypergraph models. In contrast, no much…
In this thesis, the main objects of study are probability measures on the isomorphism classes of countable, connected rooted graphs. An important class of such measures is formed by unimodular measures, which satisfy a certain equation,…
Building upon [1], this study aims to introduce fractal geometry into graph theory, and to establish a potential theoretical foundation for complex networks. Specifically, we employ the method of substitution to create and explore…
A clover is a framed trivalent graph with some additional structure, embedded in a 3-manifold. We define surgery on clovers, generalizing surgery on Y-graphs used earlier by the second author to define a new theory of finite-type invariants…