Algorithmic Experiences in Coxeter Spectral Study of P-critical Edge-Bipartite Graphs and Posets

Abstract

We study P-critical edge-bipartite graphs (bigraphs for short) Δ with n ≥ 2 vertices, by means of the nonsymmetric Gram matrix Ǧ_Δ ∈ M_n(Z), the Coxeter matrix Cox_Δ := -Ǧ_Δ · Ǧ_Δ^-tr ∈ M_n(Z), its Coxeter polynomial cox_Δ(t) = det(t · E + Ǧ_Δ · Ǧ_Δ^-tr), and its Coxeter spectrum sρecc_Δ. We recall that Δ is positive if the symmetric matrix Ǧ_Δ := Ǧ_Δ + Ǧ_Δ^tr is positive definite; and Δ is P-critical if it is not positive and each of its proper full subbigraphs is positive. It is easy to see that if two bigraphs Δ, Δ' are Z-bilinear equivalent Δ ≈_Z Δ' (i.e., there exists a matrix B ∈ Gl(n, Z) such that Ǧ_Δ = B^tr · Ǧ_Δ' · B) then their Coxeter spectra specc_Δ and specc_Δ' coincide; but the converse implication does not hold in general. One of the main questions of the Coxeter spectral analysis of connected P-critical bigraphs Δ, Δ' is whether the congruence Δ ≈_Z Δ' holds if and only if specc_Δ = specc_Δ'. In this note we discuss the problem in case when n ≤ 10 and Δ and Δ' are P-critical looρ-free bigraphs such that their Euclidean types DΔ, DΔ' ∈ {A_n, n > 1, D̃_m, m ≥ 4, Ẽ₆,Ẽ₇, Ẽ₈} coincide. In particular, we get an affirmative answer to the stated question, for a large class of P-critical bigraphs and Tits P-critical finite posets. By applying symbolic and numerical algorithms in Maple and C# we compute the set of Coxeter polynomials cox_Δ(t) for P-critical loop-free bigraphs Δ, with at most 10 vertices.