关于Corank两边二部图和简单扩展欧几里得图

Marcin Gąsiorek, D. Simson, Katarzyna Zając
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引用次数: 12

摘要

我们继续有限连通无环边二部图Δ的Coxeter谱研究,其中m+2≥3个顶点(一类有符号图),开始于[SIAM J.离散数学]。, 27(2013), 827-854],并在第12届和第13届大会上发表。本文研究了秩2的非负边二部图,即Δ的对称Gram矩阵GΔ∈Mm+2(Z)是秩m≥1的正半定。延长每一个简单的欧几里得图,m≥1,Dm, m≥4,Ẽ6Ẽ7,Ẽ8到一个顶点,我们构建一个家庭无路由循环的秩两图Am2 D̃m2,Ẽ62年,72年ẼẼ82(简称扩展欧几里德图),这样他们分类所有连接秩两个无路由循环edge-bipartite图形Δ顶点与m + 2≥3,Z-congruenceΔ~ zΔ”。这里Δ ~z Δ'表示GΔ', = Btr·GΔ·B,对于某些B∈Mm+2(z),使得det B =±1。我们提出了一种算法,使用Python中的符号和数字计算机计算,生成给定大小为m + 2≥3的所有这些边二部图,以及它们的Coxeter多项式和约简Coxeter数。此外,我们证明了对于任意有m + 2≥3个顶点的corank两连通无环边二部图Δ,存在一个简单扩展欧几里得图D,使得Δ ~z D。
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On Corank Two Edge-Bipartite Graphs and Simply Extended Euclidean Diagrams
We continue the Coxeter spectral study of finite connected loop-free edge-bipartite graphs Δ, with m+2 ≥ 3 vertices (a class of signed graphs), started in [SIAM J. Discrete Math., 27(2013), 827-854] by means of the complex Coxeter spectrum speccΔ ⊆ C and presented in our talks given in SYNASC12 and SYNASC13. Here, we study non-negative edge-bipartite graphs of corank two, in the sense that the symmetric Gram matrix GΔ ∈ Mm+2(Z) of Δ is positive semi-definite of rank m ≥ 1. Extending each of the simply laced Euclidean diagrams Am, m ≥ 1, Dm, m ≥ 4, Ẽ6, Ẽ7, Ẽ8 by one vertex, we construct a family of loop-free corank two diagrams Ãm2, D̃m2, Ẽ62, Ẽ72, Ẽ82 (called simply extended Euclidean diagrams) such that they classify all connected corank two loop-free edge-bipartite graphs Δ, with m + 2 ≥ 3 vertices, up to Z-congruence Δ ~z Δ'. Here Δ ~z Δ' means that GΔ', = Btr ·GΔ ·B, for some B ∈ Mm+2(Z) such that det B = ±1. We present algorithms that generate all such edge-bipartite graphs of a given size m + 2 ≥ 3, together with their Coxeter polynomials, and the reduced Coxeter numbers, using symbolic and numeric computer calculations in Python. Moreover, we prove that for any corank two connected loop-free edge-bipartite graph Δ, with m + 2 ≥ 3 vertices, there exists a simply extended Euclidean diagram D such that Δ ~z D.
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