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

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]. One of our aims of the paper is to study the Coxeter spectral analysis question: "Does the congruence Delta approximate to(Z) Delta' hold, for any pair of connected positive graphs Delta, Delta' is an element of RBigr(n) such that specc(Delta) = specc(Delta') and the numbers of loops in Delta and Delta' coincide?" We do it by a reduction to the Coxeter spectral study of the Gl (n, Z) D-orbits in the set Mor (D) subset of M-n (Z) of matrix morsifications of a Dynkin diagram D - D Delta is an element of uBigr(n) associated with Delta. In particular, we construct in the second part of the paper numeric algorithms for computing the connected positive edge-bipartite graphs Delta in RBigr(n), for a fixed n >= 2, mesh algorithms for computing the set of all Z-invertible matrices B is an element of Gl (n, Z) definining the Z-congruence Delta approximate to(Z) Delta', for positive graphs Delta, Delta' is an element of RBigr(n), with n >= 2 fixed, and mesh-type algorithms for the mesh root systems Gamma(R-Delta(center dot), Phi(Delta)). In the first part of the paper we present an introduction to the study of Cox-regular edge-bipartite graphs Delta with dotted loops in relation with the irreducible reduced root systems and the Dynkin diagrams B-n, n >= 2, C-n, n >= 3, F-4, G(2). Moreover, we construct a unique Phi(D)-mesh root system Gamma( R-D(center dot), Phi(D)) for each of the Cox-regular edge-bipartite graphs B-n, n >= 2, C-n, n >= 3, F-4, calG(2) of the type B-n, n >= 2, C-n, n >= 3, F-4, G(2), respectively. Our main inspiration for the study comes from the representation theory of posets, groups and algebras, Lie theory, and Diophantine geometry problems.