A Coxeter spectral classification of positive edge-bipartite graphs I. Dynkin types [formula omitted], [formula omitted], [formula omitted], [formula omitted], [formula omitted], [formula omitted], [formula omitted

Abstract

We develop a computational technique for classification of a class of signed graphs (called edge-bipartite graphs), we started in Simson (2013) [42] and Bocian et al. (2014) [6]. Here we mainly study finite positive Cox-regular edge-bipartite graphs Δ (bigraphs, for short), with n≥2 vertices, by means of the non-symmetric Gram matrix GˇΔ∈Mn(Z) defining Δ, and the complex spectrum speccΔ⊂S1:={z∈C,|z|=1} of the Coxeter matrix CoxΔ:=−GˇΔ⋅GˇΔ−tr∈Mn(Z), called the Coxeter spectrum of Δ. The bigraph Δ is said to be positive if the symmetric Gram matrix GΔ=12(GˇΔ+GˇΔtr)∈Mn(Z) is positive definite.Our aim is to classify such edge-bipartite graphs, up to the strong Gram Z-congruence Δ≈ZΔ′, where Δ≈ZΔ′ means that GˇΔ′=Btr⋅GˇΔ⋅B, for some B∈Mn(Z) with det⁡B=±1. Our main result of the paper asserts that, given a pair Δ,Δ′ of Cox-regular connected positive edge-bipartite graphs with at least one loop, there is a congruence Δ≈ZΔ′ if and only if speccΔ=speccΔ′ and det⁡GˇΔ=det⁡GˇΔ′. Moreover, given n≥2, we present a list of five types of pairwise non-congruent bigraphs such that any Cox-regular connected positive bigraph with a loop and n≥2 vertices is strongly Z-congruent with a bigraph of the list. Our main idea used in the proof is a reduction of the classification problem to the problem of computing the orbits of a finite set MorSn⊆Mn(Z) of integer matrix morsifications of the antichain Sn consisting of n vertices, with respect to the right Gram action (A,B)↦A⁎B:=Btr⋅A⋅B of the integral orthogonal group O(n,Z) on MorSn. The computational technique developed in the paper allows also to construct a symbolic algorithm that computes a matrix B∈Gl(n,Z) defining the Gram Z-congruence Δ≈ZΔ′, if it does exist.