The Antisymmetric Line Graph

Abstract

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.