Representability of Graphical Symplectic Matroids
Abstract
Chow associated with every finite multigraph G a symplectic matroid GraphSG on the signed edge set Esqcup E*, whose restrictions to transversals are frame matroids of signed graphs, and Sarvepalli asked which of these matroids are representable. For a connected graph containing a cycle, we show that every isotropic representation of S(G) admits an incidence-flow normal form. The remaining representation conditions are expressed through its admissible minors. Consequently, if GraphSG is F-representable, then the bridgeless part of G has a nowhere-zero (F,+)-flow. In characteristic other than two, GraphSG is representable if and only if every non-loop edge on a cycle has a parallel mate. In characteristic two, we characterize the loopless graphs without two vertex-disjoint cycles by the existence of a nowhere-zero (F,+)-flow, and we show that rigidly linked short cycles, and hence subdivisions of the prism, are obstructions. For $3$-connected simple graphs this gives a complete answer: GraphSG is F-representable if and only if F has characteristic two, G has no two vertex-disjoint cycles, and either |F|>2 or G≅ K5. In particular, the matroid of the Petersen graph is representable over no field. Finally, the admissible corank--nullity polynomial of an arbitrary symplectic matroid is intrinsic and satisfies a three-term contraction identity. For graphical symplectic matroids, this becomes a deletion-contraction recursion that closes up once marked vertices are allowed, and polynomials built from enveloping matroids agree with it exactly for sheltering envelopes.
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BibTeX
@misc{ganguly2026representability,
title = {{Representability of Graphical Symplectic Matroids}},
author = {Sayantan Ganguly},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01241},
eprint = {2611.01241},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01241},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Representability of Graphical Symplectic Matroids
AU - Sayantan Ganguly
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01241
UR - https://arxiv.org/abs/2611.01241
AB - Chow associated with every finite multigraph G a symplectic matroid GraphSG on the signed edge set Esqcup E^*, whose restrictions to transversals are frame matroids of signed graphs, and Sarvepalli asked which of these matroids are representable. For a connected graph containing a cycle, we show that every isotropic representation of S(G) admits an incidence-flow normal form. The remaining representation conditions are expressed through its admissible minors. Consequently, if GraphSG is F-representable, then the bridgeless part of G has a nowhere-zero (F,+)-flow. In characteristic other than two, GraphSG is representable if and only if every non-loop edge on a cycle has a parallel mate. In characteristic two, we characterize the loopless graphs without two vertex-disjoint cycles by the existence of a nowhere-zero (F,+)-flow, and we show that rigidly linked short cycles, and hence subdivisions of the prism, are obstructions. For $3$-connected simple graphs this gives a complete answer: GraphSG is F-representable if and only if F has characteristic two, G has no two vertex-disjoint cycles, and either |F|>2 or G≅ K_5. In particular, the matroid of the Petersen graph is representable over no field. Finally, the admissible corank--nullity polynomial of an arbitrary symplectic matroid is intrinsic and satisfies a three-term contraction identity. For graphical symplectic matroids, this becomes a deletion-contraction recursion that closes up once marked vertices are allowed, and polynomials built from enveloping matroids agree with it exactly for sheltering envelopes.
ER -