The Antisymmetric Line Graph
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911600080846848 |
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| author | Bal, Hartosh Singh |
| author_facet | Bal, Hartosh Singh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03087 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Antisymmetric Line Graph Bal, Hartosh Singh Combinatorics Primary 05C22, Secondary 05C50, 05C75, 05C85 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. |
| title | The Antisymmetric Line Graph |
| topic | Combinatorics Primary 05C22, Secondary 05C50, 05C75, 05C85 |
| url | https://arxiv.org/abs/2603.03087 |