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arXiv 1208.5285math.CO
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符号图的惯性集

The inertia set of a signed graph

Marina Arav, Frank J. Hall, Zhongshan Li, Hein van der Holst

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25 pages, 1 figure
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摘要：

有符号图是一对$(G,Σ)$，其中$G=(V,E)$是具有$V={1,...,n}$和$Σ\subseteq E$的图（其中允许平行边，但不允许循环）。通过$S(G,Σ)$，我们表示所有对称$V\times V$矩阵的集合$A=[a_{i,j}]$，其中如果$i$和$j$仅由偶数边连接，则$a_{i,j}<0$，如果$i$和$j$仅由奇数边连接，则$a_{i,j}>0$，如果$i$和$j$由偶数边和奇数边连接，则$a_{i,j}\in \mathbb{R}$，如果$i\not=j$和$i$和$j$不相邻，则$a_{i,j}=0$，并且$a_{i,i} \in \mathbb{R}$用于所有顶点$i$。符号图$(G,Σ)$的稳定惯性集是所有对$(p,q)$的集合，对$(p,q)$，存在一个矩阵$A\in S(G,Σ)$，其$p$为正，$q$为负，具有强Arnold性质。本文研究了（符号）图的稳定惯性集。

英文摘要：

A signed graph is a pair $(G,Σ)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V={1,...,n}$ and $Σ\subseteq E$. By $S(G,Σ)$ we denote the set of all symmetric $V\times V$ matrices $A=[a_{i,j}]$ with $a_{i,j}<0$ if $i$ and $j$ are connected by only even edges, $a_{i,j}>0$ if $i$ and $j$ are connected by only odd edges, $a_{i,j}\in \mathbb{R}$ if $i$ and $j$ are connected by both even and odd edges, $a_{i,j}=0$ if $i\not=j$ and $i$ and $j$ are non-adjacent, and $a_{i,i} \in \mathbb{R}$ for all vertices $i$. The stable inertia set of a signed graph $(G,Σ)$ is the set of all pairs $(p,q)$ for which there exists a matrix $A\in S(G,Σ)$ with $p$ positive and $q$ negative eigenvalues which has the Strong Arnold Property. In this paper, we study the stable inertia set of (signed) graphs.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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