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Inflation Agorithm for Cox-regular Postive Edge-bipartite Graphs with Loops

Makuracki B., Simson D., Zyglarski B.

Fundamenta Informaticae

2017

Vol. 153, nr 4

367--398

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 ĞΔ ∊ Mn(Z) defining Δ, its symmetric Gram matrix GΔ:=1/2[ĞΔ+ĞtrΔ]∊ Mn(1/2Z), 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 ~ZΔ' 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'↦t-aΔ' and Δ'↦t-abΔ' 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.