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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…
This article establishes that the split decomposition of graphs introduced by Cunnigham, is definable in Monadic Second-Order Logic.This result is actually an instance of a more general result covering canonical graph decompositions like…
We review the main steps of the differential equation approach to the analytic evaluation of Feynman graphs, showing at the same time its application to the 3-loop sunrise graph in a particular kinematical configuration.
Feynman diagrams are the foremost tool in the perturbative study of quantum field theory. In gauge theories, the full potential of this tool is revealed when it is combined with the Slavanov-Taylor identities associated with the local gauge…
We study the accumulation of zeros of a polynomial arising from the enumeration of edge-coloured graphs along certain limit curves. The polynomial is a variant of an edge-chromatic polynomial, which specialises to the partition function of…
We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-Hurwitz formula, study the functorial maps on Jacobians and…
Given a function $f$ in a finite field ${\mathbb F}_q$ of $q$ elements, we define the functional graph of $f$ as a directed graph on $q$ nodes labelled by the elements of ${\mathbb F}_q$ where there is an edge from $u$ to $v$ if and only if…
We study the band gap in some semi-conducting polymers with two models: H\"uckel molecular orbital theory and the so-called free electron model. The two models are directly related to spectral theory on combinatorial and metric graphs.
The aim of this paper is to generalize the notion of the coloring complex of a graph to hypergraphs. We present three different interpretations of those complexes -- a purely combinatorial one and two geometric ones. It is shown, that most…
We use mixed Hodge structures to investigate Feynman amplitudes as functions of external momenta and masses.
The article is an overview of the role of graph complexes in the Feynman path integral quantization. The underlying mathematical language is that of PROPs and operads, and their representations. The sum over histories approach, the Feynman…
An explicit algorithm is presented for testing whether two non-directed graphs are isomorphic or not. It is shown that for a graph of n vertices, the number of n independent operations needed for the test is polynomial in n. A proof that…
Graphical models represent multivariate and generally not normalized probability distributions. Computing the normalization factor, called the partition function, is the main inference challenge relevant to multiple statistical and…
Fiol, Garriga, and Yebra introduced the notion of pseudo-distance-regular vertices, which they used to develop a new characterization of distance-regular graphs. Building on that work, Fiol and Garriga developed the spectral excess theorem…
In this partly expository paper we discuss and describe some of our old and recent results on partial orders on the set (m,n)-graphs (i.e. graphs with n vertices and m edges) and some operations on graphs that are monotone with respect to…
In this paper, we generalize the notion of functional graph. Specifically, given an equation $E(X,Y) = 0$ with variables $X$ and $Y$ over a finite field $\mathbb{F}_q$ of odd characteristic, we define a digraph by choosing the elements in…
A polynomial algorithm for graphs' isomorphism testing is constructed in assumption that there exists a corresponding polynomial algorithm for graphs with trivial automorphism group.
In this article, we give a comprehensive survey of the recent progress of research on binomial edge ideal of a graph since 2018.
We investigate, using the notion of linear quotients, significative classes of connected graphs whose monomial edge ideals, not necessarily squarefree, have linear resolution, in order to compute standard algebraic invariants of the…
Recently, we proposed a new approach for calculating Feynman graphs amplitude using the Gaussian representation for propagators which was proven to be exact in the limit of graphs having an infinite number of loops. Regge behavior was also…