Related papers: On the refined counting of graphs on surfaces
The set of factorizations of permutations in to $m$ transpositions of some symmetric group $\mathcal{S}_n$ is naturally in bijection with the set of graphs of order $n$ and size $m$ with both edges and vertices labeled. We define a notion…
We define analytic maps between super Riemann surfaces which extend the notion of branched covering maps to a supersymmetric setting. We show that these super covering maps appear naturally both in symmetric product orbifolds of…
Node embedding is a key technique for representing graph nodes as vectors while preserving structural and relational properties, which enables machine learning tasks like feature extraction, clustering, and classification. While classical…
Graph embedding is a transformation of nodes of a network into a set of vectors. A good embedding should capture the underlying graph topology and structure, node-to-node relationship, and other relevant information about the graph, its…
In this paper, we introduce a method called graph fusion embedding, designed for multi-graph embedding with shared vertex sets. Under the framework of supervised learning, our method exhibits a remarkable and highly desirable synergistic…
Rigid graph theory is an active area with many open problems, especially regarding embeddings in $\mathbb{R}^d$ or other manifolds, and tight upper bounds on their number for a given number of vertices. Our premise is to relate the number…
We consider Type IIB 5-brane web diagrams for a 5D $Sp(N)$ gauge theory with an antisymmetric hypermultiplet and $N_f$ fundamental hypermultiplets. The corresponding 5-branes can be obtained by Higgsing a 5-brane web for quiver gauge…
We study biplane graphs drawn on a finite point set $S$ in the plane in general position. This is the family of geometric graphs whose vertex set is $S$ and which can be decomposed into two plane graphs. We show that every sufficiently…
We outline ideas to connect the analytic structure of Feynman amplitudes to the structure of Karen Vogtmann's {\em Outer Space}. We focus on the role of cubical chain complexes in this context, and also investigate the bordification problem…
This paper is an introduction to the language of Feynman Diagrams. We use Reshetikhin-Turaev graphical calculus to define Feynman diagrams and prove that asymptotic expansions of Gaussian integrals can be written as a sum over a suitable…
Rigidity theory studies the properties of graphs that can have rigid embeddings in a euclidean space $\mathbb{R}^d$ or on a sphere and which in addition satisfy certain edge length constraints. One of the major open problems in this field…
In the past decade, several multi-resolution representation theories for graph signals have been proposed. Bipartite filter-banks stand out as the most natural extension of time domain filter-banks, in part because perfect reconstruction,…
Expander graphs, due to their mixing properties, are useful in many algorithms and combinatorial constructions. One can produce an expander graph with high probability by taking a random graph (e.g., the union of $d$ random bijections for a…
Graph embeddings have become a key and widely used technique within the field of graph mining, proving to be successful across a broad range of domains including social, citation, transportation and biological. Graph embedding techniques…
The divisor theory of graphs views a finite connected graph $G$ as a discrete version of a Riemann surface. Divisors on $G$ are formal integral combinations of the vertices of $G$, and linear equivalence of divisors is determined by the…
A graphon is a limiting object used to describe the behaviour of large networks through a function that captures the probability of edge formation between nodes. Although the merits of graphons to describe large and unlabelled networks are…
Representation learning on graphs has been gaining attention due to its wide applicability in predicting missing links, and classifying and recommending nodes. Most embedding methods aim to preserve certain properties of the original graph…
A Heffter array is an m by n matrix with nonzero entries from Z_{2mn+1} such that i) every row and column sum to 0, and ii) no element from {x,-x} appears twice. We construct some Heffter arrays. These arrays are used to build current…
We consider surfaces embedded in a Riemannian manifold of arbitrary dimension and prove that many aspects of their differential geometry can be expressed in terms of a Poisson algebraic structure on the space of smooth functions of the…
With the rising interest in graph representation learning, a variety of approaches have been proposed to effectively capture a graph's properties. While these approaches have improved performance in graph machine learning tasks compared to…