A COXETER-GRAM CLASSIFICATION OF POSITIVE SIMPLY LACED EDGE-BIPARTITE GRAPHS

Following the spectral graph theory, a graph coloring technique, and algebraic methods in graph theory, we study the category UBigr(n) of loop-free edge-bipartite (signed) graphs Delta, with n >= 2 vertices, up to Z-congruences similar to(Z) and approximate to(Z) (defined in the paper), by means of the Gram matrix (sic)(Delta) is an element of M-n(Z), the Coxeter-Gram matrix Cox(Delta) = -(sic)(Delta) . (sic)(Delta)(-tr) is an element of M-n(Z), the spectrum specc(Delta) of Cox(Delta), the Coxeter polynomial cox Delta(t) is an element of Zleft perpendiculartright perpendicular, the Coxeter number c(Delta) >= 2, and the Z-bilinear Gram form b(Delta) : Z(n) x Z(n) -> Z of Delta. Our main inspiration for the study comes from the representation theory of posets, groups and algebras, Lie theory, and Diophantine geometry problems. One of our aims is to compute the set CGpol(n)(+) of all polynomials cox(Delta)(t), with positive connected graphs Delta in UBigr(n), for all n >= 2, and to present a framework for a classification of graphs Delta in UBigr(n), up to the congruence approximate to(Z), by means of their Coxeter polynomials cox(Delta)(t) is an element of Z[t] and Coxeter spectra. In particular, the Coxeter spectral analysis question, whether the congruence Delta approximate to(Z) Delta' holds, for any pair of connected positive graphs Delta, Delta' is an element of UBigr(n) such that specc(Delta) = specc(Delta)', is studied in the paper. One of our main results contains an affirmative answer to the question and a description of the finite set CGpol(n)(+) for n <= 8. One of the tools we apply is an inflation algorithm Delta bar right arrow D Delta that associates with any connected positive graph Delta is an element of UBigr(n) a simply laced Dynkin diagram D Delta such that Delta similar to (Z) D Delta, and cox Delta(t) and c(Delta) coincide with the Coxeter polynomial cox(A)(t) and the Coxeter number c(A) of a matrix morsification A is an element of MorD(Delta) subset of M-n(Z) for the diagram D Delta and, in case n <= 8, all pairs (cox Delta(t), c(Delta)) are listed.