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

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(R
_Δ
, Φ
_A
) of roots of Δ in the sense of [J. Pure Appl. Algebra, 215 (2010), 13-34], and matrix morsifications A ∈ Mor
_Δ
, for simply laced Dynkin diagrams Δ ∈ {A
_n
, D
_n
, E
₆
, E
₇
, E
₈
}. 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 Cox
_Δ
∈ M
_n
(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 G
_Δ
and G
_DΔ
. We also present a computer aided technique that allows us to construct a Z-congruence of the non-symmetric Gram matrices Ğ
_Δ
and Ğ
_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 ∈ {A
_n
, D
_n
, E
₆
, E
₇
}, with n ≤ 7, is obtained by a reduction to computer calculation of Gl(n, Z)
_DΔ
-orbits in the set Mor
_DΔ
, where Gl(n, Z)
_DΔ
is the isotropy group of the Dynkin diagram DΔ.