On Computing Non-negative Loop-Free Edge-Bipartite Graphs

Abstract

We continue the Coxeter spectral study of finite connected loop-free edge-bipartite graphs Δ, with n ≥ 2 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. Here, we discuss Coxeter spectral analysis problems of non-negative edge-bipartite graphs of corank s ≤ n-1, which means that the symmetric Gram matrix G_Δ ∈ M_n(ℤ) is positive semi-definite of rank n-s ≤ n. In particular, we study in details the loop-free edge-bipartite graphs of corank s = n - 1. We present algorithms that generate all such edge-bipartite graphs of a given size and, using symbolic and numerical computer calculations in Python, and we obtain their complete classification in relation with Diophantine geometry problems. We also construct algorithms that allow us to classify all connected loop-free non-negative edge-bipartite graphs Δ, with a fixed number n ≥ 2 of vertices, by means of their Coxeter spectra specc_Δ.