2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing 2013

DOI: 10.1109/synasc.2013.16

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On Computing Non-negative Loop-Free Edge-Bipartite Graphs

Grzegorz Marczak

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Katarzyna Zając

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Cited by 4 publications

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Applications of mesh algorithms and self-dual mesh geometries of root Coxeter orbits to a Horn-Sergeichuk type problem

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2022

Linear Algebra and its Applications

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Applications of mesh algorithms and self-dual mesh geometries of root Coxeter orbits to a Horn-Sergeichuk type problem

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2022

Linear Algebra and its Applications

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On Coxeter Type Classification of Loop-Free Edge-Bipartite Graphs and Matrix Morsifications

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2013

2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

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We continue and complete a Coxeter spectral study (presented in our talk given in SYNASC11 and SYNASC12) of the root systems in the sense of Bourbaki, the mesh geometries Γ(RΔ, ΦA) of roots of Δ in the sense of [J. Pure Appl. Algebra, 215 (2010), 13-34], and matrix morsifications A ∈ MorΔ, for simply laced Dynkin diagrams Δ ∈ {An, Dn, E6, E7, E8}. Here we report on algorithmic and morsification technique for the Coxeter spectral analysis of connected loop-free edge-bipartite graphs Δ with n ≥ 2 vertices by means of the Coxeter matrix CoxΔ ∈ Mn(Z), the Coxeter spectrum specc Δ , and an inflation algorithm associating to any connected loop-free positive bigraph Δ a simply laced Dynkin diagram DΔ, and defining a Z-congruence of the symmetric Gram matrices GΔ and GDΔ. We also present a computer aided technique that allows us to construct a Z-congruence of the non-symmetric Gram matricesǦΔ andǦ Δ , if the Coxeter spectra specc Δ and specc Δ coincide. A complete Coxeter spectral classification of positive edge-bipartite graphs Δ of Coxeter-Dynkin types DΔ ∈ {An, Dn, E6, E7}, with n ≤ 7, is obtained by a reduction to computer calculation of Gl(n, Z)DΔ-orbits in the set MorDΔ, where Gl(n, Z)DΔ is the isotropy group of the Dynkin diagram DΔ.

On Corank Two Edge-Bipartite Graphs and Simply Extended Euclidean Diagrams

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2014

2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

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