Product Line Sigraphs

Intuitively, a signed graph is a graph in which every edge is labeled with a + or - sign. For each edge, its sign represents the mode of the relationship between the vertices it joins. In a signed graph, cycles can be naturally given the sign corresponding to the product of its edges. Then, a signed graph is called balanced when all the cycles have positive sign. Balanced signed graph have multiple applications in the field of social networks. Consequently, there is a significant amount of research in the problem of determining if a signed graph is balanced or not. In particular, some authors investigated extensions to signed graph of the line graph and studied under what circumstances the signed graphs obtained are balanced or not. This paper presents a new operation, which is also an extension to signed graphs of the line graph, with the property that applied to any signed graph always produces a balanced signed graph.