Symbolic computation of strong Gram congruences for Cox-regular positive edge-bipartite graphs with loops

Abstract

The paper can be viewed as a second part of the author's paper (Simson (2018) [43]). Our main aim is to construct symbolic algorithms for the Coxeter spectral classification of a class of signed graphs (called edge-bipartite graphs), with n≥2 vertices, we started in Simson (2013) [37] and Bocian et al. (2014) [6]. More precisely, we construct algorithms for the classification (up to the strong Gram Z-congruence Δ≈ZΔ′) of all finite connected Cox-regular edge-bipartite graphs Δ with at least one loop (bigraphs, for short) that are positive in the sense that the associated symmetric Gram matrix GΔ=12(GˇΔ+GˇΔtr)∈Mn(12Z) is positive definite, where GˇΔ∈Mn(Z) is the non-symmetric Gram matrix of Δ defined in Section 1 and Δ≈ZΔ′ means that there is a Z-invertible matrix B∈Mn(Z) such that GˇΔ′=Btr⋅GˇΔ⋅B. We recall from [43] that every such a bigraph Δ, with a loop, is strongly Gram Z-congruent with a bigraph DΔ, that is one of the positive Cox-regular bigraphs Bn, n≥2, Cn, n≥3, F4, M4, G2 presented in Section 1. Here, by applying the geometry of mesh root system technique, we construct symbolic algorithms that compute the correspondence Δ↦DΔ and the set GlˇΔ(n,Z)DΔ of all Z-invertible matrices B∈Mn(Z) defining the strong Gram Z-congruence Δ≈ZDΔ, for any connected positive bigraph Δ, with n≥2 vertices and at least one loop.