On mesh geometries of root Coxeter orbits and mesh algorithms for corank two edge-bipartite signed graphs

Abstract

Following a Coxeter spectral analysis problems for positive edge-bipartite graphs (signed multigraphs with a separation property) introduced in Simson (2013) [41] and Simson (2013) [42], we study analogous problems for loop-free corank two edge-bipartite graphs Δ=(Δ0,Δ1), i.e., for edge-bipartite graphs Δ, with n+2=|Δ0|≥3 vertices such that its rational symmetric Gram matrix GΔ:=12(GˇΔ+GˇΔtr)∈Mn+2(Q) is positive semi-definite of rank n. We study such connected edge-bipartite signed graphs (bigraphs, for short), up to the strong Gram congruence Δ≈ZΔ′, by means of the non-symmetric Gram matrix GˇΔ∈Mn+2(Z), the Coxeter matrix CoxΔ:=−GˇΔ⋅GˇΔ−tr∈Mn+2(Z), its complex spectrum speccΔ⊆C, and an associated simply laced Dynkin diagram DynΔ∈{An,Dn,E6,E7,E8}, with n≥1 vertices. It is known that if the congruence holds Δ≈ZΔ′ (i.e., there exists B∈Mn+2(Z) such that det⁡B=±1 and GˇΔ′=Btr⋅GˇΔ⋅B) then (speccΔ,DynΔ)=(speccΔ′,DynΔ′).The inverse implication is proved in [Fund. Inform. 152 (2017) 185–222] for all pairs of non-negative corank two bigraphs Δ, Δ′ without loops with n+2≤6 vertices. One of the main aims of the paper is to find algorithms that construct the set Cong(Δ,Δ′) of all Z-invertible matrices B∈Mn+2(Z) defining the congruence Δ≈ZΔ′, for any pair of corank two bigraphs Δ and Δ′ such that (speccΔ,DynΔ)=(speccΔ′,DynΔ′). We do it in case when Δ and Δ′ have at most six vertices in three steps: (i) to any such Δ, we associate algorithmically a ΦΔ-mesh translation quiver ΓΔ•=Γ(RˆΔ,ΦΔ) in Zn+2 (called ΦΔ-mesh geometry of roots of Δ, in the sense of Simson (2013) [42]), where ΦΔ:Zn+2→Zn+2, v↦v⋅CoxΔ, is the Coxeter transformation of Δ; (ii) we define an injective contravariant map h˜•:Cong(Δ,Δ′)→Iso(ΓΔ′•,ΓΔ•), where Iso(ΓΔ′•,ΓΔ•) is the set of all mesh translation quiver isomorphisms h˜B:ΓΔ′•→ΓΔ•, and (iii) a numerical algorithm computing a required matrix B∈Cong(Δ,Δ′) is constructed by means of the shape of the ΦΔ-mesh translation quiver ΓΔ•, its toroidal-tubular structure, and an analysis of admissible isomorphisms h˜B:ΓΔ′•→ΓΔ•.