
TL;DR
This paper introduces a new perspective on the classical signed line graph, using its invariants to analyze properties of the underlying graph, including bipartization and frustration indices, with spectral bounds and optimization characterizations.
Contribution
It establishes that the switching class of the antisymmetric line graph determines the original graph up to isomorphism, linking frustration index to bipartization parameters and providing spectral bounds.
Findings
The switching class of the antisymmetric line graph determines the original graph up to isomorphism.
The frustration index relates to classical bipartization parameters with specific bounds.
An exact optimization identity links frustration index to a Boolean Laplacian problem.
Abstract
Let be a finite simple graph with oriented incidence matrix . The signed graph on edge set 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 determines up to isomorphism modulo isolated vertices, and we relate the frustration index to classical bipartization parameters. In particular, we show \[ \operatorname{def}(G)\le \ell(\mathcal A(G))\le (\Delta(G)-1)\operatorname{def}(G), \] 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…
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