最小特征值至少为\(-3\)的符号图及其格
Signed graphs with fixed smallest eigenvalue at least $-3$ and their lattices
浏览论文内容
中文总结 AI 辅助
研究最小特征值至少为\(-3 - \varepsilon\)的连通符号图，证明最小度足够大时最小特征值至少为\(-3\)，相关格是标准格\(\mathbb{Z}^n\)与\(E_8\)根格直和的子格，还讨论了源于无根不可约幺模格的最小特征值为\(-3\)的符号图。
中文摘要 AI 辅助
在本文中，我们考虑最小特征值至少为\(-3 - \varepsilon\)（\(\varepsilon\)为小的正实数）的连通符号图。我们证明，如果这样的符号图具有足够大的最小度，那么其最小特征值至少为\(-3\)，并且与之相关的由平方范数为\(3\)的向量生成的格是标准格\(\mathbb{Z}^n\)和根格\(E_8\)的副本的直和的子格。此外，存在无限多个最小特征值至少为\(-3\)的连通符号图，它们以其作为真诱导子图。此外，我们讨论了由无根不可约幺模格产生的最小特征值为\(-3\)的符号图。
英文摘要
In this paper, we consider connected signed graphs with smallest eigenvalue at least $-3-\varepsilon$ for a small positive constant $\varepsilon$. We prove that if such a signed graph has sufficiently large minimum valency, then its smallest eigenvalue is at least $-3$, and the lattice associated with it, which is generated by squared norm $3$ vectors, is a sublattice of a direct sum of the standard lattice $\mathbb{Z}^n$ and copies of the root lattice $E_8$. Moreover, there exist infinitely many connected signed graphs with smallest eigenvalue at least $-3$ containing it as a proper induced subgraph. Furthermore, we discuss signed graphs with smallest eigenvalue $-3$ arising from rootless irreducible unimodular lattices.
