Related papers: On the refined counting of graphs on surfaces
Determining the number of embeddings of Laman graph frameworks is an open problem which corresponds to understanding the solutions of the resulting systems of equations. In this paper we investigate the bounds which can be obtained from the…
Whitney proved that 3-connected planar graphs admit a unique embedding on the sphere. In contrast, Enami investigated embeddings of 3-connected cubic planar graphs on non-spherical surfaces with non-negative Euler characteristic. He…
Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. Topological drawings of complete graphs and of complete bipartite graphs have been…
We study maps from a 2D world-sheet to a 2D target space which include folds. The geometry of folds is discussed and a metric on the space of folded maps is written down. We show that the latter is not invariant under area preserving…
We develop a general diagrammatic theory of welded graphs, and provide an extension of Satoh's Tube map from welded graphs to ribbon surface-links. As a topological application, we obtain a complete link-homotopy classification of so-called…
Bi-partite ribbon graphs arise in organising the large $N$ expansion of correlators in random matrix models and in the enumeration of observables in random tensor models. There is an algebra $\mathcal{K}(n)$, with basis given by bi-partite…
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.
Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface…
Two types of connected chord diagrams with chord endpoints lying in a collection of ordered and oriented real segments are considered here: the real segments may contain additional bivalent vertices in one model but not in the other. In the…
We provide formulas for invariants defined on a tensor product of defining representations of unitary groups, under the action of the product group. This situation has a physical interpretation, as it is related to the quantum mechanical…
Graph embedding is a powerful method in parallel computing that maps a guest network $G$ into a host network $H$. The performance of an embedding can be evaluated by certain parameters, such as the dilation, the edge congestion and the…
We study the number of connected graphs with $n$ vertices that cannot be written as the cartesian product of two graphs with fewer vertices. We give an upper bound which implies that for large $n$ almost all graphs are both connected and…
The spectrum of chiral operators in supersymmetric quiver gauge theories is typically much larger in the free limit, where the superpotential terms vanish. We find that the finite N counting of operators in any free quiver theory, with a…
We associate all small subgraph counting problems with a systematic graph encoding/representation system which makes a coherent use of graphlet structures. The system can serve as a unified foundation for studying and connecting many…
In a recent article J. Phys. Compl. 4 (2023) 035005, Kawamoto evoked statistical physics methods for the problem of counting graphs with a prescribed degree sequence. This treatment involved truncating a particular Taylor expansion at the…
In this note we elaborate on some notions of surface area for discrete graphs which are closely related to the inverse degree. These notions then naturally lead to associated connectivity measures of graphs and to the definition of a…
There are several different extensions of the Tutte polynomial to graphs embedded in surfaces. To help frame the different options, here we consider the problem of extending the Tutte polynomial to cellularly embedded graphs starting from…
Learning efficient graph representation is the key to favorably addressing downstream tasks on graphs, such as node or graph property prediction. Given the non-Euclidean structural property of graphs, preserving the original graph data's…
In this review, we discuss recent developments concerning efficient calculations of multi-loop multi-leg scattering amplitudes. Inspired by the remarkable properties of the Loop-Tree Duality (LTD), we explain how to reconstruct an integrand…
We propose an algorithm to estimate the topology of an embedded metric graph from a well-sampled finite subset of the underlying graph.