A Coxeter spectral classification of positive edge-bipartite graphs. I: Dynkin types \(\mathcal{B}_n\), \(\mathcal{C}_n\), \(\mathcal{F}_4\), \(\mathcal{G}_2\), \(\mathbb{E}_6\), \(\mathbb{E}_7\), \(\mathbb{E}_8\)

DOI10.1016/J.LAA.2018.07.013zbMath1396.05049OpenAlexW2883133809MaRDI QIDQ1790462

Daniel Simson

Publication date: 2 October 2018

Published in: Linear Algebra and its Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.laa.2018.07.013

zbMATH Keywords

Coxeter spectrumintegral quadratic formedge-bipartite graphinflation algorithmquasi-Cartan matrixDynkin graphGram congruence

Mathematics Subject Classification ID

Symbolic computation and algebraic computation (68W30) Combinatorics in computer science (68R05) Quadratic forms over general fields (11E04) Graphs and linear algebra (matrices, eigenvalues, etc.) (05C50) Quadratic and bilinear forms, inner products (15A63) Signed and weighted graphs (05C22)

Related Items (16)

Weyl roots and equivalences of integral quadratic forms ⋮ Coxeter Invariants for Non-negative Unit Forms of Dynkin Type 𝔸r ⋮ On algorithmic Coxeter spectral analysis of positive posets ⋮ Weyl orbits of matrix morsifications and a Coxeter spectral classification of positive signed graphs and quasi-Cartan matrices of Dynkin type \(\mathbb{A}_n\) ⋮ Symbolic computation of strong Gram congruences for Cox-regular positive edge-bipartite graphs with loops ⋮ On polynomial time inflation algorithm for loop-free non-negative edge-bipartite graphs ⋮ A Strong Gram Classification of Non-negative Unit Forms of Dynkin Type 𝔸r ⋮ Quadratic algorithm to compute the Dynkin type of a positive definite quasi-Cartan matrix ⋮ On mesh geometries of root Coxeter orbits and mesh algorithms for corank two edge-bipartite signed graphs ⋮ A Coxeter spectral classification of positive edge-bipartite graphs. II: Dynkin type \(\mathbb{D}_n\) ⋮ Applications of mesh algorithms and self-dual mesh geometries of root Coxeter orbits to a Horn-Sergeichuk type problem ⋮ On the structure of loop-free non-negative edge-bipartite graphs ⋮ Root systems and inflations of non-negative quasi-Cartan matrices ⋮ A computational technique in Coxeter spectral study of symmetrizable integer Cartan matrices ⋮ Congruence of rational matrices defined by an integer matrix ⋮ A Graph Theoretical Framework for the Strong Gram Classification of Non-negative Unit Forms of Dynkin Type 𝔸n

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• A Gram classification of non-negative corank-two loop-free edge-bipartite graphs
• Applications of matrix morsifications to Coxeter spectral study of loop-free edge-bipartite graphs
• Inflation algorithm for loop-free non-negative edge-bipartite graphs of corank at least two
• \(P\)-critical integral quadratic forms and positive unit forms: an algorithmic approach
• Mesh geometries of root orbits of integral quadratic forms
• Coxeter energy of graphs
• Algebras whose Coxeter polynomials are products of cyclotomic polynomials.
• Tubes in derived categories and cyclotomic factors of the Coxeter polynomial of an algebra.
• Tame algebras and integral quadratic forms
• Periodicity in bilinear lattices and the Coxeter formalism
• On weakly positive unit forms
• Signed graphs
• Canonical matrices for linear matrix problems
• A Gram classification of principal Cox-regular edge-bipartite graphs via inflation algorithm
• Graphical characterization of positive definite non symmetric quasi-Cartan matrices
• Characterization of clusterability of signed graph in terms of Newcomb's balance of sentiments
• Systems of subspaces of a unitary space
• The Dynkin type of a non-negative unit form.
• On Coxeter type study of non-negative posets using matrix morsifications and isotropy groups of Dynkin and Euclidean diagrams.
• Numeric and mesh algorithms for the Coxeter spectral study of positive edge-bipartite graphs and their isotropy groups
• Periodic Coxeter matrices and their associated quadratic forms
• On the notion of balance of a signed graph
• Mesh Algorithms for Coxeter Spectral Classification of Cox-regular Edge-bipartite Graphs with Loops, I. Mesh Root Systems
• Mesh Algorithms for Coxeter Spectral Classification of Cox-regular Edge-bipartite Graphs with Loops, II. Application to Coxeter Spectral Analysis
• Algorithms for Isotropy Groups of Cox-regular Edge-bipartite Graphs
• On Algorithmic Study of Non-negative Posets of Corank at Most Two and their Coxeter-Dynkin Types
• A Framework for Coxeter Spectral Analysis of Edge-bipartite Graphs, their Rational Morsifications and Mesh Geometries of Root Orbits
• A Coxeter--Gram Classification of Positive Simply Laced Edge-Bipartite Graphs
• Mesh Algorithms for Solving Principal Diophantine Equations, Sand-glass Tubes and Tori of Roots
• A New Algorithm for On-line Coloring Bipartite Graphs
• Graph Theoretical and Algorithmic Characterizations of Positive Definite Symmetric Quasi-Cartan Matrices
• Numeric Algorithms for Corank Two Edge-bipartite Graphs and their Mesh Geometries of Roots
• Inflation Agorithm for Cox-regular Postive Edge-bipartite Graphs with Loops
• Cubic Algorithm to Compute the Dynkin Type of a Positive Definite Quasi-Cartan Matrix
• Congruences of Edge-bipartite Graphs with Applications to Grothendieck Group Recognition I. Inflation Algorithm Revisited*
• Congruences of Edge-bipartite Graphs with Applications to Grothendieck Group Recognition II. Coxeter Type Study*
• Symbolic Algorithms Computing Gram Congruences in the Coxeter Spectral Classification of Edge-bipartite Graphs, I. A Gram Classification
• Symbolic Algorithms Computing Gram Congruences in the Coxeter Spectral Classification of Edge-bipartite Graphs, II. Isotropy Mini-groups
• Algorithms Determining Matrix Morsifications, Weyl orbits, Coxeter Polynomials and Mesh Geometries of Roots for Dynkin Diagrams
• CLUSTER ALGEBRAS OF FINITE TYPE AND POSITIVE SYMMETRIZABLE MATRICES
• A characterization of positive unit forms. II

This page was built for publication: A Coxeter spectral classification of positive edge-bipartite graphs. I: Dynkin types \(\mathcal{B}_n\), \(\mathcal{C}_n\), \(\mathcal{F}_4\), \(\mathcal{G}_2\), \(\mathbb{E}_6\), \(\mathbb{E}_7\), \(\mathbb{E}_8\)