Inflation algorithm for loop-free non-negative edge-bipartite graphs of corank at least two

Abstract

We continue the study of finite connected loop-free edge-bipartite graphs Δ, with m≥3 vertices (a class of signed graphs), we started in Simson (2013) [48] and M. Gąsiorek et al. (2016) [19] by means of the non-symmetric Gram matrix GˇΔ∈Mm(Z) of Δ, its symmetric Gram matrix GΔ:=12[GˇΔ+GˇΔtr]∈Mm(12Z), and the Gram quadratic form qΔ:Zm→Z. In particular, we study connected non-negative edge-bipartite graphs Δ, with n+r≥3 vertices of corank r≥2, in the sense that the symmetric Gram matrix GΔ∈Mn+r(Z) of Δ is positive semi-definite of rank n≥1. The edge-bipartite graphs Δ of corank r≥2 are studied, up to the weak Gram Z-congruence Δ∼ZΔ′, where Δ∼ZΔ′ means that GΔ′=Btr⋅GΔ⋅B, for some B∈Mn+r(Z) such that det⁡B=±1. Our main result of the paper asserts that, given a connected edge-bipartite graph Δ with n+r≥3 vertices of corank r≥2, there exists a suitably chosen sequence t⁎− of the inflation operators of the form Δ′↦tab−Δ′ such that the composite operator Δ↦t⁎−Δ reduces Δ to a connected bigraph Dˆn(r) such that Δ∼ZDˆn(r) and Dˆn(r) is one of the canonical r-vertex extensions Aˆn(r), n≥1, Dˆn(r), n≥4, Eˆ6(r), Eˆ7(r), and Eˆ8(r), with n+r vertices, of the simply laced Dynkin diagrams An,Dn,E6,E7,E8, with n≥1 vertices. The algorithm constructs also a matrix B∈Mn+r(Z), with det⁡B=±1, defining the weak Gram Z-congruence Δ∼ZDˆn(r), that is, satisfying the equation GDˆn(r)=Btr⋅GΔ⋅B.