edge is satisfied. In physics this corresponds to a ground state (a state of lowest energy), if such a state exists. An important problem in physics, especially in the theory of spin glasses, is to determine a state with the fewest frustrated edges. Related concepts Gain graphs used in topological graph theory as a means to construct graph embeddings in surfaces are known as "voltage graphs" (Gross 1974; Gross and Tucker 1977). The term "gain graph" is more usual in other contexts, e.g., biased graph theory and matroid theory. The term group-labelled graph has also been used, but it is ambiguous since "group labels" may be—and have been—treated as weights. Since much of the theory of gain graphs is a special case of that of biased graphs (and much of the theory of biased graphs is a generalization of that of gain graphs), the reader should refer to the article on biased graphs for more information and examples. References Jonathan L. Gross (1974), Voltage graphs. Discrete Mathematics, Vol. 9, pp. 239–246. J.L. Gross and T.W. Tucker (1977), Generating all graph coverings by permutation voltage assignments. Discrete Mathematics, Vol. 18, pp. 273–283. Thomas Zaslavsky (1989), Biased graphs. I. Bias, balance, and gains. Journal of Combinatorial Theory, Series B, Vol. 47, 32–52. Thomas Zaslavsky (1991), Biased graphs. II. The three matroids. Journal of Combinatorial Theory, Series B, Vol. 51, 46–72. Thomas Zaslavsky (1999). A mathematical bibliography of signed and gain graphs and allied areas. Electronic Journal of Combinatorics, Dynamic Surveys in Combinatorics, #DS8. Thomas Zaslavsky (2003), Biased graphs IV: Geometrical realizations. Journal of Combinatorial Theory, Series B, Vol. 89, no. 2, pp. 231–297. Matroid theory Extensions and generalizations of graphs