A Coding Approach to Signed Graphs

The cocycle code of an undirected graph $\Gamma$ is the linear span over F2 of the characteristic vectors of cutsets. (If $\Gamma$ is complete bipartite, this is the generalized Gale-Berlekamp code.) The natural bijection between the cosets of this code and the switching classes of signed graphs based on $\Gamma$ is used to show that the number of such classes is equal to the number of even-degree subgraphs of $\Gamma$ in both the labeled and unlabeled cases and to improve by coding theory previous bounds on $D(T)$, the maximum line index of imbalance of signings of $\Gamma$. Bounds on $D(T)$ are obtained in terms of the genus $\Gamma$ and on the number of unlabeled even-degree subgraphs in terms of $D(T)$. Numerous examples are treated, including the "grid" (or "lattice" graphs that are of interest in the Ising model of spin glasses.