On Algorithmic Study of Non-negative Posets of Corank at Most Two and their Coxeter-Dynkin Types

摘要

Following the Coxeter spectral analysis of loop-free edge-bipartite graphs. and finite posets I, with n >= 2 vertices, introduced and developed in [SIAM J. Discrete Math., 27(2013), 827-854], we present a Coxeter spectral classification of finite posets I, with n >= 2 elements. Here we study the connected posets I that are non-negative of corank one or two, in the sense that the symmetric Gram matrix 1/2 (CI + C-I(tr)) is an element of M-n (Q) is positive semi-definite of corank one or two, where C-I is an element of M-n (Z) is the incidence matrix of I. We study such posets I by means of the Dynkin type Dyn(I) and the Coxeter polynomial cox(I)(t) : = det (t center dot E Cox(I)) is an element of Z [t], where Cox(I) : = -C-I center dot C-I(-tr) is an element of M-n (Z) is the Coxeter matrix of I.