Applications of matrix morsifications to Coxeter spectral study of loop-free edge-bipartite graphs

Following the spectral graph theory and a Coxeter type technique applied in the representation theory of groups, algebras, coalgebras, and posets, we study the category UBigrnUBigrn of connected loop-free edge-bipartite graphs ΔΔ with n≥2n≥2 vertices (a class of signed graphs) by means of their Coxeter spectrum speccΔ⊂C, i.e., the spectrum of the Coxeter matrix CoxΔ:=−ǦΔ⋅ǦΔ∈Mn(Z)(1.2), where ǦΔ∈Mn(Z) is the non-symmetric Gram matrix (1.1) of ΔΔ. One of our aims is to compute the set CGpoln+ of all polynomials coxΔ(t), with positive graphs ΔΔ in UBigrnUBigrn, for all n≥2n≥2. Moreover, we study the following Coxeter spectral analysis problem (1.4): “Does the  ZZ-congruence  Δ≈ZΔ′Δ≈ZΔ′hold (i.e. the matrices  ǦΔand  ǦΔ′are  ZZ-congruent), for any pair of connected positive loop-free edge-bipartite graphs  ΔΔ,  Δ′Δ′such that  speccΔ=speccΔ′”  . We show that our problems reduce to the study of the set MorD⊆Mn(Z) of matrix morsifications and their mesh root systems, for the simply laced Dynkin diagrams AnAn, DnDn, E6E6, E7E7, E8E8. On this way we construct algorithms allowing us to compute the finite set CGpoln+ and we get a solution of the problem (1.4) for n≤9n≤9 and Δ,Δ′Δ,Δ′ of types AnAn and DnDn.