Mesh Algorithms for Coxeter Spectral Classification of Cox-regular Edge-bipartite Graphs with Loops,II. Application to Coxeter Spectral Analysis

Abstract

Abstract

This is the second 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 $𝒰ℬig{r}_n$ of loop-free edge-bipartite (signed) graphs Δ, with n ≥ 2 vertices, we study here the larger category $ℛℬig{r}_n$ of Cox-regular edge-bipartite graphs Δ (possibly with dotted loops), up to the usual ℤ-congruences ~_Z and ≈_Z. The positive graphs Δ in $ℛℬig{r}_n$, with dotted loops, are studied by means of the complex Coxeter spectrum $spec{c}_{\Delta }\subset ℂ$, the irreducible mesh root systems of Dynkin types ${𝔹}_n,n\ge 2,{ℂ}_n,n\ge 3,{𝔽}_{\text{4}},{𝔾}_{\text{2}}$, the isotropy group Gl(n, ℤ)_Δ (containing the Weyl group of Δ), 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 our two part paper is to study the Coxeter spectral analysis question: “Does the congruence Δ ≈_ℤ Δ′ hold, for any pair of connected positive graphs $\Delta ,{\Delta }^{\prime }\in ℛℬig{r}_n$ such that $\text{spec}{\text{c}}_{\Delta }=\text{spec}{\text{c}}_{{\Delta }^{\prime }}$ and the numbers of loops in Δ and Δ′ coincide?” We do it by a reduction to the Coxeter spectral study of the $\text{G1}(n,\mathbb{Z}{)}_D$-orbits in the set $Mo{r}_D\subset {𝕄}_n(\mathbb{Z})$ of matrix morsifications of a Dynkin diagram $D=D\Delta \in 𝒰Big{r}_n$ associated with Δ. In this second part, we construct numeric algorithms for computing the connected positive edge-bipartite graphs Δ in $ℛℬig{r}_n$, for a fixed n ≥ 2, mesh algorithms for computing the set of all ℤ-invertible matrices B ∈ $\text{G1}(n,\mathbb{Z})$ definining the ℤ-congruence $\Delta {\approx }_{\mathbb{Z}}{\Delta }^{\text{'}}$, for positive graphs $\Delta ,{\Delta }^{\prime }\in ℛℬig{r}_n$, with ngeq2 fixed, and mesh-type algorithms for the mesh root systems $\Gamma (R_{\Delta }^{•},{\Phi }_{\Delta })$. We also present a classification and a structure type results for positive Cox-regular edge-bipartite graphs Δ with dotted loops.