A Gram classification of non-negative corank-two loop-free edge-bipartite graphs

We continue the Coxeter spectral study of finite connected loop-free edge-bipartite graphs Δ, with m+2≥3m+2≥3 vertices (a class of signed graphs), started in Simson (2013) [49], by means of the non-symmetric Gram matrix GˇΔ∈Mm+2(Z) of Δ, its symmetric Gram matrix GΔ:=12[GˇΔ+GˇΔtr]∈Mm+2(12Z), the Gram quadratic form qΔ:Zm+2→ZqΔ:Zm+2→Z, and the Coxeter spectrum speccΔ⊂CspeccΔ⊂C, i.e., the complex spectrum of the Coxeter matrix CoxΔ:=−GˇΔ⋅GˇΔ−tr∈Gl(m+2,Z). In the present paper we study non-negative edge-bipartite graphs of corank two, in the sense that the symmetric Gram matrix GΔ∈Mm+2(Z)GΔ∈Mm+2(Z) of Δ is positive semi-definite of rank m≥1m≥1. One of our aims is to get a complete classification of all connected corank-two loop-free edge-bipartite graphs Δ, with m+2≥3m+2≥3 vertices, up to the weak Gram ZZ-congruence Δ∼ZΔ′Δ∼ZΔ′, where Δ∼ZΔ′Δ∼ZΔ′ means that GΔ′=Btr⋅GΔ⋅BGΔ′=Btr⋅GΔ⋅B, for some B∈Mm+2(Z)B∈Mm+2(Z) such that det⁡B=±1det⁡B=±1. By one-vertex extensions of the simply laced Euclidean diagrams A˜m, m≥1m≥1, D˜m, m≥4m≥4, E˜6,E˜7,E˜8, we construct a family of connected loop-free corank-two diagrams A˜m(2),D˜m(2),E˜6(2),E˜7(2),E˜8(2) (called simply extended Euclidean diagrams) such that they classify all connected corank-two loop-free edge-bipartite graphs Δ, with m+2≥3m+2≥3 vertices, up to the weak Gram ZZ-congruence Δ∼ZΔ′Δ∼ZΔ′. A structure of connected corank-two loop-free edge-bipartite graphs Δ is described. It is shown that every such Δ contains a connected positive edge-bipartite subgraph Δ′Δ′, that is ZZ-congruent with a simply laced Dynkin diagram DynΔDynΔ (called the Dynkin type of Δ) such that Δ is a two-point extension Δ′[[u,w]]Δ′[[u,w]] of Δ′Δ′ along two roots u,wu,w of the positive definite Gram form qΔ′:Zm→ZqΔ′:Zm→Z. This yields a combinatorial algorithm (Δ′,u,w)↦Δ′[[u,w]](Δ′,u,w)↦Δ′[[u,w]] allowing us to construct all connected corank-two loop-free edge-bipartite graphs Δ, with m+2≥3m+2≥3 vertices and D=DynΔD=DynΔ, from the triples (Δ′,u,w)(Δ′,u,w), where Δ′Δ′ is positive of the Dynkin type D  , and u,wu,w are roots of the positive definite Gram form qΔ′:Zm→ZqΔ′:Zm→Z.