交错族与有向图的Hermitian谱范数
Interlacing families and the Hermitian spectral norm of digraphs
摘要：
证明了对任意有限连通图$G$，存在$G$的一个定向，使得其对应的Hermite邻接矩阵的谱半径小于或等于$G$的泛覆盖的谱半径（等式当且仅当$G$是树）.这解决了Mohar提出的一个问题。证明使用的方法交错家庭的多项式，是由马库斯，斯皮尔曼，和Srivastava在他们的开创性工作存在的无限家庭拉马努金图。
英文摘要：
It is proved that for any finite connected graph $G$, there exists an orientation of $G$ such that the spectral radius of the corresponding Hermitian adjacency matrix is smaller or equal to the spectral radius of the universal cover of $G$ (with equality if and only if $G$ is a tree). This resolves a problem proposed by Mohar. The proof uses the method of interlacing families of polynomials that was developed by Marcus, Spielman, and Srivastava in their seminal work on the existence of infinite families of Ramanujan graphs.
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