A signed graph is a graph whose edges are labeled by signs. This is a bibliography of signed graphs and related mathematics. Several kinds of labelled graph have been called "signed" yet are mathematically very different. I distinguish four types: Group-signed graphs: the edge labels are elements of a 2-element group and are multiplied around a polygon (or along any walk). Among the natural generalizations are larger groups and vertex signs. Sign-colored graphs, in which the edges are labelled from a two-element set that is acted upon by the sign group: - interchanges labels, + leaves them unchanged. This is the kind of "signed graph" found in knot theory. The natural generalization is to more colors and more general groups —  or no group. Weighted graphs, in which the edge labels are the elements +1 and -1 of the integers or another additive domain. Weights behave like numbers, not signs; thus I regard work on weighted graphs as outside the scope of the bibliography — ex cept (to some extent) when the author calls the weights "signs". Labelled graphs where the labels have no structure or properties but are called "signs" for any or no reason.

/pdf/a-mathematical-bibliography-of-signed-and-gain-graphs-and-3a49f10hs2.pdf

A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas

The Leray-Schauder degree is defined for mappings of the form $I-C$, where
$C$ is a compact mapping from the closure of an open bounded subset of a
Banach space $X$ into $X$. Since the fifties, a lot of work has been devoted
in extending this theory to the same type of mappings on some nonlinear
spaces, and in extending the class of mappings in the frame of Banach spaces
or manifolds. New applications of Leray-Schauder theory and its extensions
have also been given, specially in bifurcation theory, nonlinear boundary
value problems and equations in ordered spaces. The paper surveys those
developments.

https://projecteuclid.org/download/pdf_1/euclid.tmna/1475179840

Leray-Schauder degree: a half century of extensions and applications

We survey recent results on the representation theory of symplectic reflection algebras, focusing particularly on connections with symplectic quotient singularities and their resolutions, spaces of representations of quivers, and on category O.

/pdf/symplectic-reflection-algebras-wo7m24un4j.pdf

Symplectic reflection algebras

In this paper we analyze the vibrations of nonlinear structures by means of the novel approach of isogeometric finite elements. The fundamental idea of isogeometric finite elements is to apply the same functions, namely B-Splines and NURBS (Non-Uniform Rational B-Splines), for describing the geometry and for representing the numerical solution. In case of linear vibrational analysis, this approach has already been shown to possess substantial advantages over classical finite elements, and we extend it here to a nonlinear framework based on the harmonic balance principle. As application, the straight nonlinear Euler–Bernoulli beam is used, and overall, it is demonstrated that isogeometric finite elements with B-Splines in combination with the harmonic balance method are a powerful means for the analysis of nonlinear structural vibrations. In particular, the smoother k-method provides higher accuracy than the p-method for isogeometric nonlinear vibration analysis.

/pdf/isogeometric-analysis-of-nonlinear-euler-bernoulli-beam-1rjg01cp90.pdf

Isogeometric analysis of nonlinear Euler–Bernoulli beam vibrations

Study of wave propagation in strongly nonlinear periodic lattices using a harmonic balance approach

The series is devoted to the publication of high-level monographs which cover the whole spectrum of current nonlinear analysis and applications in various fields, such as optimization, control theory, systems theory, mechanics, engineering, and other sciences. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of nonlinear analysis. Contributions which are on the borderline of nonlinear analysis and related fields and which stimulate further research at the crossroads of these areas are particularly welcome. Editor-in-Chief Jurgen Appell, Wurzburg, Germany Honorary and Advisory Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Umberto Mosco, Worcester, Massachusetts, USA Louis Nirenberg, New York, USA Alfonso Vignoli, Rome, Italy Editorial Board Manuel del Pino, Bath, UK, and Santiago, Chile Mikio Kato, Nagano, Japan Wojciech Kryszewski, Torun, Poland Vicentiu D. Radulescu, Krakow, Poland Simeon Reich, Haifa, Israel Please submit book proposals to Jurgen Appell. Titles in planning include Lucio Damascelli and Filomena Pacella, Morse Index of Solutions of Nonlinear Elliptic Equations (2019) Tomasz W. Dlotko and Yejuan Wang, Critical Parabolic-Type Problems (2019) Rafael Ortega, Periodic Differential Equations in the Plane: A Topological Perspective (2019) Ireneo Peral Alonso and Fernando Soria, Elliptic and Parabolic Equations Involving the Hardy-Leray Potential (2020) Cyril Tintarev, Profile Decompositions and Cocompactness: Functional-Analytic Theory of Concentration Compactness (2020) Takashi Suzuki, Semilinear Elliptic Equations: Classical and Modern Theories (2021)

Approximation Procedures in Nonlinear Oscillation Theory

Generalizing the classical matrix-tree theorem we provide a for- mula counting subgraphs of a given graph with a fixed 2-core. We use this gen- eralization to obtain an analog of the matrix-tree theorem for the root system Dn (the classical theorem corresponds to the An-case). Several byproducts of the developed technique, such as a new formula for a specialization of the multivariate Tutte polynomial, are of independent interest.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2006/0013/0005/MRL-2006-0013-0005-a007.pdf

Around matrix-tree theorem

https://www.hse.ru/data/2014/12/09/1317707374/LAA.pdf

Matrix-tree theorems and discrete path integration

Consider factorizations into transpositions of an n-cycle in the symmetric group S"n. To every such factorization we assign a monomial in variables w"i"j that retains the transpositions used, but forgets their order. Summing over all possible factorizations of n-cycles we obtain a polynomial that happens to admit a closed expression. From this expression we deduce a formula for the number of 1-faced embeddings of a given graph.

/pdf/cycle-factorizations-and-1-faced-graph-embeddings-45g4u9k7si.pdf

Cycle factorizations and 1-faced graph embeddings

We introduce and study deformations of finite-dimensional modules over rational Cherednik algebras. Our main tool is a generalization of usual harmonic polynomials for Coxeter groups -- the so-called quasiharmonic polynomials. A surprising application of this approach is the construction of canonical elementary symmetric polynomials and their deformations for all Coxeter groups.

Yurii Burman

Papers

Approximation Procedures in Nonlinear Oscillation Theory

Around matrix-tree theorem

Matrix-tree theorems and discrete path integration

Cycle factorizations and 1-faced graph embeddings

Quasiharmonic polynomials for Coxeter groups and representations of Cherednik algebras