论文标题
重新审视的Elser核和
The Elser nuclei sum revisited
论文作者
论文摘要
修复有限的无向图$γ$和$γ$的顶点$ v $。令$ e $为$γ$的边缘。如果$γ$的每个边缘具有至少一个可以通过$ f $ path的$ V $连接到$ v $的$ f $ f $ $ e $ pandemic的子集$ f $ f $ f $ f $ f $ e $ e $。 1984年，埃尔瑟（Elser）表明$ \ left（-1 \ right）^{\ left |的总和所有大流行子集的$ f $ $ e $ $ e $的f \ right |} $如果$ e \ e \ neq \ varnothing $。我们通过反向签名的相互作用给出了简单的证明，并讨论变体，概括和改进，揭示了与抽象凸度（抗抗原剂的概念）和离散摩尔斯理论的联系。
Fix a finite undirected graph $Γ$ and a vertex $v$ of $Γ$. Let $E$ be the set of edges of $Γ$. We call a subset $F$ of $E$ pandemic if each edge of $Γ$ has at least one endpoint that can be connected to $v$ by an $F$-path (i.e., a path using edges from $F$ only). In 1984, Elser showed that the sum of $\left(-1\right)^{\left| F\right|}$ over all pandemic subsets $F$ of $E$ is $0$ if $E\neq \varnothing$. We give a simple proof of this result via a sign-reversing involution, and discuss variants, generalizations and refinements, revealing connections to abstract convexity (the notion of an antimatroid) and discrete Morse theory.
