A Framework for Coxeter Spectral Analysis of Edge-bipartite Graphs, their Rational Morsifications and Mesh Geometries of Root Orbits
Following the spectral Coxeter analysis of matrix morsifications for Dynkin diagrams, the spectral graph theory, a graph coloring technique, and algebraic methods in graph theory, we continue our study of the category UBigr(n) of loop-free edge-bipartite (signed) graphs Delta, with n >= 2 vertices, by means of the Coxeter number c(Delta), the Coxeter spectrum specc(Delta) of Delta, that is, the spectrum of the Coxeter polynomial cox(Delta)(t) is an element of Z [t] and the Z-bilinear Gram form b(Delta) : Z(n) x Z(n) -> Z of Delta [SIAM J. Discrete Math. 27(2013)]. Our main inspiration for the study comes from the representation theory of posets, groups and algebras, Lie theory, and Diophantine geometry problems. We show that the Coxeter spectral classification of connected edge-bipartite graphs Delta in UBigr(n) reduces to the Coxeter spectral classification of rational matrix morsifications A is an element of (M) over cap or (D Delta). for a simply-laced Dynkin diagram D Delta associated with Delta. Given Delta in UBigr(n), we study the isotropy subgroup Gl(n, Z)(Delta) of Gl(n, Z) that contains the Weyl group W-Delta and acts on the set (M) over cap or(Delta) of rational matrix morsifications A of Delta in such a way that the map A bar right arrow (specc(A), det A, c(Delta)) is Gl(n, Z)(Delta)-invariant. It is shown that, for n <= 6, specc(Delta) is the spectrum of one of the Coxeter polynomials listed in Tables 3.11-3.11(a) (we determine them by computer search using symbolic and numeric computation). The question, if two connected positive edge-bipartite graphs Delta,Delta' in UBigr n, with specc(Delta) = specc(Delta'), are Z-bilinear equivalent, is studied in the paper. The problem if any Z-invertible matrix A is an element of M-n (Z) is Z-congruent with its transpose A(tr) is also discussed.
edge-bipartite graph mesh geometry of root orbits matrix morsification dynkin diagram coxeter spectrum weyl group toroidal mesh algorithm inflation algorithm euler bilinear form sand-glass tube