反对称线图
The Antisymmetric Line Graph
AI总结：
本文研究了反对称线图的开关类作为无向图不变量的来源，证明其在孤立顶点模下确定图的同构性，并关联了摩擦指数与经典双分参数，推导了摩擦指数的精确优化公式。
摘要：
设G为有限简单图，其定向接触矩阵为D。在边集E(G)上的符号图，其邻接矩阵为$A_{\mathcal A(G)}=D^T D-2I$，在符号线图文献中是经典的。本文研究其标准开关类作为无向图不变量的来源。证明开关类$\mathcal A(G)$在孤立顶点模下确定G的同构性，并关联摩擦指数$\ell(\mathcal A(G))$与经典双分参数。特别地，证明$\operatorname{def}(G)\le \ell(\mathcal A(G))\le (\Delta(G)-1)\operatorname{def}(G)$，且对于立方图，$\ell(\mathcal A(G))=2\\,\operatorname{oct}(G)$。然后证明精确优化恒等式$\ell(\mathcal A(G)) =\frac{1}{4}\sum_{v\in V(G)} d(v)^2 -\frac{1}{4}\max_{x\in\{\pm1\}^{E(G)}}\\|Dx\\|^2$，因此$\ell(\mathcal A(G))$恰好是布尔边空间拉普拉斯优化问题。这给出了拉普拉斯最大特征值的谱下界，立方谱下界在奇圈转置中的下界，以及显式家族比较，显示谱界和缺陷界支配不同的领域：在奇圈上谱界是渐近无意义的，而在完全多部图上它已经精确捕捉了$\ell(\mathcal A(G))$的3/4真实值。因此本文用经典的符号线图以新方式：作为普通图组合不变量的来源，特别是通过摩擦和奇圈转置现象。
英文摘要：
Let $G$ be a finite simple graph with oriented incidence matrix $D$. The signed graph on edge set $E(G)$ with adjacency matrix \[ A_{\mathcal A(G)}=D^{\mathsf T}D-2I \] is classical in the signed-line-graph literature. In this paper we study its canonical switching class as a source of invariants of the underlying unsigned graph. We prove that the switching class of $\mathcal A(G)$ determines $G$ up to isomorphism modulo isolated vertices, and we relate the frustration index $\ell(\mathcal A(G))$ to classical bipartization parameters. In particular, we show \[ \operatorname{def}(G)\le \ell(\mathcal A(G))\le (Δ(G)-1)\operatorname{def}(G), \] and, for cubic graphs, \[ \ell(\mathcal A(G))=2\,\operatorname{oct}(G). \] We then prove the exact optimization identity \[ \ell(\mathcal A(G)) = \frac14\sum_{v\in V(G)} d(v)^2 -\frac14\max_{x\in\{\pm1\}^{E(G)}}\|Dx\|^2, \] so $\ell(\mathcal A(G))$ is exactly a Boolean edge-space Laplacian optimization problem. This yields a spectral lower bound in terms of the largest Laplacian eigenvalue, a cubic spectral lower bound on odd cycle transversal, and explicit family-level comparisons showing that the spectral and defect bounds govern different regimes: on odd cycles the spectral bound is asymptotically vacuous, while on complete multipartite graphs it already captures exactly $3/4$ of the true value of $\ell(\mathcal A(G))$. Thus the paper uses a classical signed line graph in a new way: as a source of combinatorial invariants of ordinary graphs, especially through frustration and odd-cycle-transversal phenomena.
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