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• W3101264650 abstract "We continue the Coxeter spectral study of finite positive edge-bipartite signed (multi)graphs Δ (bigraphs, for short), with n≥2 vertices started in Simson (2013) [44] and developed in Simson (2018) [49]. We do it by means of the non-symmetric Gram matrix GˇΔ∈Mn(Z) defining Δ, its Gram quadratic form qΔ:Zn→Z, v↦v⋅GˇΔ⋅vtr (that is positive definite, by definition), the complex spectrum speccΔ⊂S1:={z∈C,|z|=1} of the Coxeter matrix CoxΔ:=−GˇΔ⋅GˇΔ−tr∈Mn(Z), called the Coxeter spectrum of Δ, and the Coxeter polynomial coxΔ(t):=det⁡(t⋅E−CoxΔ)∈Z[t]. One of the aims of the Coxeter spectral analysis is to classify the connected bigraphs Δ with n≥2 vertices up to the ℓ-weak Gram Z-congruence Δ∼ℓZΔ′ and up to the strong Gram Z-congruence Δ≈ZΔ′, where Δ∼ℓZΔ′ (resp. Δ≈ZΔ′) means that det⁡GˇΔ=det⁡GˇΔ′ and GΔ′=Btr⋅GΔ⋅B (resp. GˇΔ′=Btr⋅GˇΔ⋅B), for some B∈Mn(Z) with det⁡B=±1, where GΔ:=12[GˇΔ+GˇΔtr]∈Mn(12Z). Here we study connected signed simple graphs Δ, with n≥2 vertices, that are positive, i.e., the symmetric Gram matrix GΔ∈Mn(12Z) of Δ is positive definite. It is known that every such a signed graph is ℓ-weak Gram Z-congruent with a unique simply laced Dynkin graph DynΔ∈{An,Dn,n≥4,E6,E7,E8}, called the Dynkin type of Δ. A classification up to the strong Gram Z-congruence Δ≈ZΔ′ is still an open problem and only partial results are known. In this paper, we obtain such a classification for the positive signed simple graphs Δ of Dynkin type Dn by means of the family of the signed graphs Dn(1)=Dn,Dn(2),…,Dn(rn) constructed in Section 2, for any n≥4, where rn=⌞n/2⌟. More precisely, we prove that any connected signed simple graph of Dynkin type Dn, with n≥4 vertices, is strongly Z-congruent with a signed graph Dn(s), for some s≤rn, and its Coxeter polynomial coxΔ(t) is of the form (ts+1)(tn−s+1). We do it by a matrix morsification type reduction to the classification of the conjugacy classes in the integral orthogonal group O(n,Z) of the integer orthogonal matrices C, with det⁡(E−C)=4." @default.
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• W3101264650 date "2021-03-01" @default.
• W3101264650 modified "2023-10-14" @default.
• W3101264650 title "A Coxeter spectral classification of positive edge-bipartite graphs II. Dynkin type <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si1.svg><mml:msub><mml:mrow><mml:mi mathvariant=double-struck>D</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>" @default.
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• W3101264650 doi "https://doi.org/10.1016/j.laa.2020.11.001" @default.
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