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

Abstract

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.