Symbolic Algorithms Computing Gram Congruences in the Coxeter Spectral Classification of Edge-bipartite Graphs, I. A Gram Classification

Abstract

We continue the Coxeter spectral study of the category UBigrm of loop-free edgebipartite (signed) graphs ∆, with m ≥ 2 vertices, we started in [SIAM J. Discr. Math. 27(2013), 827-854] for corank r = 0 and r = 1. Here we study the class of all non-negative edge-bipartite graphs ∆ ∈ UBigrn+r of corank r ≥ 0, up to a pair of the Gram Z-congruences ∼Z and ≈Z, by means of the non-symmetric Gram matrix Ǧ∆ ∈ Mn+r(Z) of ∆, the symmetric Gram matrix G∆ := 1 2 [Ǧ∆ + Ǧ tr ∆ ] ∈ Mn+r(Z), the Coxeter matrix Cox∆ := −Ǧ∆ · Ǧ −tr ∆ ∈ Mn+r(Z) and its spectrum specc∆ ⊂ C, called the Coxeter spectrum of ∆. One of the aims in the study of the category UBigrn+r is to classify the equivalence classes of the non-negative edge-bipartite graphs in UBigrn+r with respect to each of the Gram congruences ∼Z and ≈Z. In particular, the Coxeter spectral analysis question, when the strong congruence ∆ ≈Z ∆′ holds (hence also ∆ ∼Z ∆′ holds), for a pair of connected non-negative graphs ∆,∆′ ∈ UBigrn+r such that specc∆ = specc∆′ , is studied in the paper. One of our main aims is an algorithmic description of a matrix B defining the Gram Z-congruences ∆ ≈Z ∆′ and ∆ ∼Z ∆′, that is, a Z-invertible matrix B ∈Mn+r(Z) such that Ǧ∆′ = B · Ǧ∆ ·B and G∆′ = B ·G∆ ·B, respectively. We show that, given a connected non-negative edge-bipartite graph ∆ in UBigrn+r of corank r ≥ 0 there exists a simply laced Dynkin diagram D, with n vertices, and a connected canonical r-vertex extension D := D of D of corank r (constructed in Section 2) such that ∆ ∼Z D. We also show that every matrix B defining the strong Gram Zcongruence ∆ ≈Z ∆′ in UBigrn+r has the form B = C∆ · B · C−1 ∆′ , where C∆, C∆′ ∈ Mn+r(Z) are fixed Z-invertible matrices defining the weak Gram congruences ∆ ∼Z D and ∆′ ∼Z D with an r-vertex extended graph D, respectively, and B ∈ Mn+r(Z) is Z-invertible matrix lying in the isotropy group Gl(n+r,Z)D of D. Moreover, each of the columns κ ∈ Z n+r of B is a root of ∆, Address for correspondence: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Torun, Poland. ∗Supported by Polish Research Grant NCN 2011/03/B/ST1/00824. Received November 2015; revised March 2016 20 D. Simson / Symbolic Algorithms Computing the Gram Congruence, I i.e., κ · Ǧ∆ · κ = 1. Algorithms constructing the set of all such matrices B are presented in case when r = 0. We essentially use our construction of a morsification reduction map φD : UBigrD → MorD that reduces (up to ≈Z) the study of the set UBigrD of all connected non-negative edgebipartite graphs ∆ in UBigrn+r such that ∆ ∼Z D to the study of Gl(n+r,Z)D-orbits in the set MorD ⊆ Gl(n+r,Z) of all matrix morsifications of the graph D.