Affiliations: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland. {bartmak,simson,bzstyler}@mat.umk.pl
Correspondence:
[*]
Address for correspondece: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland.
Abstract: We continue the study of finite connected edge-bipartite graphs Δ, with m ≥ 2 vertices (a class of signed graphs), started in [SIAM J. Discrete Math. 27(2013), 827-854] and developed in [Fund. Inform. 139(2015), 249-275, 145(2016), 19-48] by means of the non-symmetric Gram matrix G∨Δ∈𝕄n(ℤ) defining Δ, its symmetric Gram matrix GΔ:=12[GΔ∨+GΔtr∨]∈𝕄n(12ℤ), and the Gram quadratic form qΔ : ℤn → ℤ. In the present paper we study connected positive Cox-regular edge-bipartite graphs Δ, with n ≥ 2 vertices, in the sense that the symmetric Gram matrix GΔ ∈ 𝕄n(ℤ) of Δ is positive definite. Our aim is to classify such Cox-regular edge-bipartite graphs with at least one loop by means of an inflation algorithm, up to the weak Gram ℤ-congruence Δ ~ℤ Δ′, where Δ ~ℤ Δ′ means that GΔ′ = Btr · GΔ · B, for some B ∈ 𝕄n(ℤ) such that det B = ±1. Our main result of the paper asserts that, given a positive connected Cox-regular edge-bipartite graph Δ with n ≥ 2 vertices and with at least one loop there exists a Cox-regular edge-bipartite Dynkin graph 𝒟n ∨ {ℬn, 𝒞n, ℱ4, 𝒢2} with loops and a suitably chosen sequence t•− of the inflation operators of one of the types Δ′↦ta−Δ′
and Δ′↦tab−Δ′ such that the composite operator Δ↦t•−Δ
reduces Δ to the bigraph 𝒟n such that Δ ~ℤ 𝒟n and the bigraphs Δ, 𝒟n have the same number of loops. The algorithm does not change loops and the number of vertices, and computes a matrix B ∈ 𝕄n(ℤ), with det B = ±1, defining the weak Gram ℤ-congruence Δ ~ℤ 𝒟n, that is, satisfying the equation G𝒟n = Btr · GΔ · B.
Keywords: edge-bipartite graph, Dynkin graph, inflation algorithm, Gram matrix, unit quadratic form, Coxeter spectrum, isotropy group
