Mesh Algorithms for Coxeter Spectral Classification of Cox-regular Edge-bipartite Graphs with Loops, I. Mesh Root Systems

摘要

This is the first part of our two part paper with the same title. Following our Coxeter spectral study in [Fund. Inform. [123(2013), 447-490] and [SIAM J. Discr. Math. 27(2013), 827-854] of the category UBigr(n) of loop-free edge-bipartite (signed) graphs Delta, with n >= 2 vertices, we study here the larger category RBigr(n) of Cox-regular edge-bipartite graphs Delta (possibly with dotted loops), up to the usual Z-congruences similar to(Z) and approximate to(Z). The positive graphs Delta in RBigr(n), with dotted loops, are studied by means of the complex Coxeter spectrum specc(Delta) subset of C, the irreducible mesh root systems of Dynkin types B-n, n >= 2, C-n, n >= 3, F-4, G(2), the isotropy group Gl (n, Z)Delta (containing the Weyl group of Delta), and by applying the matrix morsification technique introduced in [J. Pure A ppl. Algebra 215(2011), 13-24] and [Fund. Inform. [123(2013), 447-490].