Numeric and mesh algorithms for the Coxeter spectral study of positive edge-bipartite graphs and their isotropy groups

Extended Abstract 1 Preliminaries. Following the spectral graph theory, a graph coloring technique and algebraic methods in graph theory (see [5], [13]), we continue a Coxeter spectral study the category Bigrn of connected non-negative loop-free edge-bipartite (signed) graphs ∆, with n ≥ 1 vertices (bigraphs, in short), and their morsifications introduced by the second named author in [22]-[25]. We study the bigraphs ∆ by means of the Coxeter spectra specc∆ ⊆ S1 := {z ∈ C; |z| = 1}, the integer non-symmetric Gram matrix Ǧ∆ ∈ Mn(Z) and the Coxeter-Gram matrix Cox∆ := −Ǧ∆ · Ǧ−tr ∆ ∈ Mn(Z). One of the main aims of the Coxeter spectral analysis we develop here, is to study positive loop-free bigraphs ∆ in Bigrn, up to the Z-bilinear congruence ∆ ≈Z ∆′, by means of the Coxeter spectrum specc∆ ⊆ S1, the Coxeter transformation Φ∆ : Zn → Zn and the Coxeter number c∆. In particular, we study the following Coxeter spectral analysis problem stated in [25]: (CSAP) Show that, given a pair of connected positive loop-free bigraphs ∆ and ∆′ in Bigrn, the equality specc∆ = specc∆′ implies ∆ ≈Z ∆′, that is, implies the existence of a Z-invertible matrix B ∈ Mn(Z) such that Ǧ∆′ = Ǧ∆ ∗ B = Btr · Ǧ∆ ·B. Our main idea is to reduce the problem to the Coxeter spectral analysis of the Weyl orbits of matrix morsifications for the simply-laced Dynkin diagrams: An : •1−−−−•2−−−−•3−−−− . . . −−−−•−−−−•n E6 : •4 | •1−−−−•2−−−−•3−−−−•5−−−−•6;