A Coxeter spectral classification of positive edge-bipartite graphs I. Dynkin types Bn, Cn, F4, G2, E6, E7, E8

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.