# Linear codes over signed graphs

## TL;DR

This paper establishes formulas connecting graph invariants to the parameters of linear codes derived from signed graphs, and explores algebraic properties related to circuits, cocircuits, and frustration index.

## Contribution

It provides new formulas for code parameters and algebraic invariants of signed graphs, linking graph theory and coding theory in novel ways.

## Key findings

- Formulas for minimum distance and Hamming weights of codes from signed graphs
- Determination of regularity of ideals of circuits and cocircuits
- Algebraic formula for frustration index in signed graphs

## 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.