On Corank Two Edge-Bipartite Graphs and Simply Extended Euclidean Diagrams

Abstract

We continue the Coxeter spectral study of finite connected loop-free edge-bipartite graphs Δ, with m+2 ≥ 3 vertices (a class of signed graphs), started in [SIAM J. Discrete Math., 27(2013), 827-854] by means of the complex Coxeter spectrum specc_Δ ⊆ C and presented in our talks given in SYNASC12 and SYNASC13. Here, we study non-negative edge-bipartite graphs of corank two, in the sense that the symmetric Gram matrix G_Δ ∈ M_m+2(Z) of Δ is positive semi-definite of rank m ≥ 1. Extending each of the simply laced Euclidean diagrams A_m, m ≥ 1, D_m, m ≥ 4, Ẽ₆, Ẽ₇, Ẽ₈ by one vertex, we construct a family of loop-free corank two diagrams Ã_m², D̃_m², Ẽ₆², Ẽ₇², Ẽ₈² (called simply extended Euclidean diagrams) such that they classify all connected corank two loop-free edge-bipartite graphs Δ, with m + 2 ≥ 3 vertices, up to Z-congruence Δ ~z Δ'. Here Δ ~z Δ' means that G_Δ', = B^tr ·G_Δ ·B, for some B ∈ M_m+2(Z) such that det B = ±1. We present algorithms that generate all such edge-bipartite graphs of a given size m + 2 ≥ 3, together with their Coxeter polynomials, and the reduced Coxeter numbers, using symbolic and numeric computer calculations in Python. Moreover, we prove that for any corank two connected loop-free edge-bipartite graph Δ, with m + 2 ≥ 3 vertices, there exists a simply extended Euclidean diagram D such that Δ ~z D.