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 UBigrn U B i g r n of connected loop-free edge-bipartite graphs ? ? with n?2 n ? 2 vertices (a class of signed graphs) by means of their Coxeter spectrum specc ? ? C , i.e., the spectrum of the Coxeter matrix Cox ? : = ? G ? ? ? G ? ? ? M n ( Z ) (1.2), where G ? ? ? M n ( Z ) is the non-symmetric Gram matrix (1.1) of ? ? . One of our aims is to compute the set C G p o l n + of all polynomials cox ? ( t ) , with positive graphs ? ? in UBigrn U B i g r n , for all n?2 n ? 2 . Moreover, we study the following Coxeter spectral analysis problem (1.4): Does the Z Z -congruence ??Z?? ? ? Z ? ? hold (i.e. the matrices  G ? ? and  G ? ? ? are Z Z -congruent), for any pair of connected positive loop-free edge-bipartite graphs ? ? , ?? ? ? such that  specc ? = specc ? ?  . We show that our problems reduce to the study of the set Mor D ? M n ( Z ) of matrix morsifications and their mesh root systems, for the simply laced Dynkin diagrams An A n , Dn D n , E6 E 6 , E7 E 7 , E8 E 8 . On this way we construct algorithms allowing us to compute the finite set C G p o l n + and we get a solution of the problem (1.4) for n?9 n ? 9 and ?,?? ? , ? ? of types An A n and Dn D n .
edge-bipartite graph coxeter matrix coxeter spectral analysis combinatorial algorithms computer algebra