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

摘要

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 ?(R?, ?A) of roots of ? in the sense of [J. Pure Appl. Algebra, 215 (2010), 13 -- 34], and matrix morsifications A ? Mor?, for simply laced Dynkin diagrams ? ? {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 ? with n ? 2 vertices by means of the Coxeter matrix Cox? Mn (Z), the Coxeter spectrum specc?, and an inflation algorithm associating to any connected loop-free positive bigraph Δ, a simply laced Dynkin diagram Δ, and defining a Z-congruence of the symmetric Gram matrices Δ and Δ. We also present a computer aided technique that allows us to construct a Z-congruence of the non-symmetric Gram matrices of δ and δ, if the Coxeter spectra coincide. A complete Coxeter spectral classification of positive edge-bipartite graphs of Coxeter-Dynkin types An, Dn, E6, E7, with n