# Linear codes over signed graphs

• José Martínez-Bernal
• Miguel A. Valencia-Bucio
• Rafael H. Villarreal
Article

## Abstract

We give formulas, in terms of graph theoretical invariants, for the minimum distance and the generalized Hamming weights of the linear code generated by the rows of the incidence matrix of a signed graph over a finite field, and for those of its dual code. Then we determine the regularity of the ideals of circuits and cocircuits of a signed graph, and prove an algebraic formula in terms of the multiplicity for the frustration index of an unbalanced signed graph.

## Keywords

Generalized Hamming weight Incidence matrix Linear code Signed graph Vector matroid Edge connectivity Frustration index Circuit Cycle Regularity Multiplicity

## Mathematics Subject Classification

Primary 94B05 Secondary 94C15 05C40 05C22 13P25

## Notes

### Acknowledgements

We thank Thomas Zaslavsky for suggesting to generalize our work on incidence matrix codes of graphs to signed graphs, and for pointing out that the edge biparticity of a graph is a special case of the frustration index of a signed graph. We thank the referee for a careful reading of the paper and for the improvements suggested. Computations with Macaulay2 [14], Matroids [8], and SageMath [30] were important to verifying and computing examples given in this paper.

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