On the Isomorphism Problem for Coxeter Groups and Related Topics

2014 ◽  
pp. 217-238
Author(s):  
Koji Nuida
Keyword(s):  
Coxeter Groups ◽  
Isomorphism Problem

Cohomology of Groups

2019 ◽  
pp. 213-326
Author(s):  
Graham Ellis
Keyword(s):  
Finite Groups ◽  
Coxeter Groups ◽  
Nilpotent Groups ◽  
Isomorphism Problem ◽  
Serre Spectral Sequence ◽  
Homological Perturbation Theory ◽  
Mathieu Groups ◽  
Transfer Homomorphism ◽  
Homological Perturbation ◽  
Cohomology Of Groups

This chapter introduces the basic ingredients of the cohomology of groups and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: integral homology of finite groups such as the Mathieu groups, homology of crystallographic groups, homology of nilpotent groups, homology of Coxeter groups, transfer homomorphism, homological perturbation theory, mod-p comology rings of small finite p-groups, Lyndon-Hocshild-Serre spectral sequence, Bokstein operation, Steenrod squares, Stiefel-Whitney classes, Lie algebras, the modular isomorphism problem, and Bredon homology.

scholarly journals The isomorphism problem for k-trees is complete for logspace

2012 ◽  
Vol 217 ◽  
pp. 1-11 ◽  
Author(s):  
V. Arvind ◽  
Bireswar Das ◽  
Johannes Köbler ◽  
Sebastian Kuhnert
Keyword(s):  
Isomorphism Problem

scholarly journals MET: a Java package for fast molecule equivalence testing

2020 ◽  
Vol 12 (1) ◽  
Author(s):  
Jördis-Ann Schüler ◽  
Steffen Rechner ◽  
Matthias Müller-Hannemann
Keyword(s):  
Chemical Properties ◽  
Graph Isomorphism ◽  
Isomorphism Problem ◽  
Equivalence Classes ◽  
Equivalence Testing ◽  
Equivalence Test ◽  
Graph Isomorphism Problem ◽  
2D Structure ◽  
Node Labels ◽  
Labelled Graphs

AbstractAn important task in cheminformatics is to test whether two molecules are equivalent with respect to their 2D structure. Mathematically, this amounts to solving the graph isomorphism problem for labelled graphs. In this paper, we present an approach which exploits chemical properties and the local neighbourhood of atoms to define highly distinctive node labels. These characteristic labels are the key for clever partitioning molecules into molecule equivalence classes and an effective equivalence test. Based on extensive computational experiments, we show that our algorithm is significantly faster than existing implementations within , and . We provide our Java implementation as an easy-to-use, open-source package (via GitHub) which is compatible with . It fully supports the distinction of different isotopes and molecules with radicals.

scholarly journals Automorphisms of odd Coxeter groups

2021 ◽  
Author(s):  
Tushar Kanta Naik ◽  
Mahender Singh
Keyword(s):  
Coxeter Groups

scholarly journals Isomorphism, canonization, and definability for graphs of bounded rank width

2021 ◽  
Vol 64 (5) ◽  
pp. 98-105
Author(s):  
Martin Grohe ◽  
Daniel Neuen
Keyword(s):  
Fixed Point ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Graph Isomorphism ◽  
Isomorphism Problem ◽  
Descriptive Complexity ◽  
Graph Isomorphism Problem ◽  
Structural Graph ◽  
Graph Classes ◽  
Bounded Rank

We investigate the interplay between the graph isomorphism problem, logical definability, and structural graph theory on a rich family of dense graph classes: graph classes of bounded rank width. We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3 k + 4) is a complete isomorphism test for the class of all graphs of rank width at most k. A consequence of our result is the first polynomial time canonization algorithm for graphs of bounded rank width. Our second main result addresses an open problem in descriptive complexity theory: we show that fixed-point logic with counting expresses precisely the polynomial time properties of graphs of bounded rank width.

scholarly journals Diamond Subgraphs in the Reduction Graph of a One-Rule String Rewriting System

2021 ◽  
Vol 178 (3) ◽  
pp. 173-185
Author(s):  
Arthur Adinayev ◽  
Itamar Stein
Keyword(s):  
Isomorphism Problem ◽  
Complete Characterization ◽  
Hasse Diagram ◽  
Subgraph Isomorphism ◽  
Rewriting Systems ◽  
Rewriting System ◽  
String Rewriting ◽  
Reduction Graph

In this paper, we study a certain case of a subgraph isomorphism problem. We consider the Hasse diagram of the lattice Mk (the unique lattice with k + 2 elements and one anti-chain of length k) and find the maximal k for which it is isomorphic to a subgraph of the reduction graph of a given one-rule string rewriting system. We obtain a complete characterization for this problem and show that there is a dichotomy. There are one-rule string rewriting systems for which the maximal such k is 2 and there are cases where there is no maximum. No other intermediate option is possible.

scholarly journals The 300 “correlators” suggests 4D, $$ \mathcal{N} $$ = 1 SUSY is a solution to a set of Sudoku puzzles

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Aleksander J. Cianciara ◽  
S. James Gates ◽  
Yangrui Hu ◽  
Renée Kirk
Keyword(s):  
Coxeter Groups ◽  
Auxiliary Field ◽  
Permutation Groups ◽  
Weight Space ◽  
Field Problem ◽  
Truncated Octahedron ◽  
Color Problem ◽  
Minimal Representations

Abstract A conjecture is made that the weight space for 4D, $$ \mathcal{N} $$ N -extended supersymmetrical representations is embedded within the permutahedra associated with permutation groups 𝕊d. Adinkras and Coxeter Groups associated with minimal representations of 4D, $$ \mathcal{N} $$ N = 1 supersymmetry provide evidence supporting this conjecture. It is shown that the appearance of the mathematics of 4D, $$ \mathcal{N} $$ N = 1 minimal off-shell supersymmetry representations is equivalent to solving a four color problem on the truncated octahedron. This observation suggest an entirely new way to approach the off-shell SUSY auxiliary field problem based on IT algorithms probing the properties of 𝕊d.

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