The critical exponent：一种新的图不变量
The critical exponent: a novel graph invariant
摘要：
FitzGerald和Horn（1977）的一个令人惊讶的结果表明，$A^{\circ α} := (a_{ij}^α)$是半正定的（p.s.d.）对于每个条目非负$n \times n$ p.s.d.矩阵$A = (a_{ij})$当且仅当$α$是正整数或$α\geq n-2$。给定一个图$G$，我们考虑的细化问题的特征集$\mathcal{H}_G$的条目的权力保持积极的矩阵与零模式编码的$G$。使用代数和组合的方法，我们研究了几何的$G$如何影响集$\mathcal{H}_G$。我们的治疗提供了新的和令人兴奋的组合数学和分析之间的连接，并导致我们引入和计算一个新的图形不变量称为临界指数。
英文摘要：
A surprising result of FitzGerald and Horn (1977) shows that $A^{\circ α} := (a_{ij}^α)$ is positive semidefinite (p.s.d.) for every entrywise nonnegative $n \times n$ p.s.d. matrix $A = (a_{ij})$ if and only if $α$ is a positive integer or $α\geq n-2$. Given a graph $G$, we consider the refined problem of characterizing the set $\mathcal{H}_G$ of entrywise powers preserving positivity for matrices with a zero pattern encoded by $G$. Using algebraic and combinatorial methods, we study how the geometry of $G$ influences the set $\mathcal{H}_G$. Our treatment provides new and exciting connections between combinatorics and analysis, and leads us to introduce and compute a new graph invariant called the critical exponent.
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