Product Line Sigraphs. Daniela Ferrero Department of Mathematics Texas State University San Marcos, TX U.S.A. Abstract Intuitively, a signed graph is a graph in which ev- ery edge is labeled with a + or - sign. For each edge, its sign represents the mode of the relation- ship 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 signif- icant amount of research in the problem of de- termining if a signed graph is balanced or not. In particular, some authors investigated exten- sions to signed graph of the line graph and stud- ied 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. Keywords: signed graphs, balance, line graph 1. Introduction Signed graphs or sigraphs, for short, are simple graphs (i.e. without loops or multiple edges) in which the edges are labeled with a sign, + or -. The motivation for studying sigraphs arises in the field of social sciences. For example, psy- chologists often model the sociometric structure of a group of people as a square matrix with el- ements -1, 0 and 1 representing disliking, indif- ference and liking, respectively. When a matrix of this sort is symmetric, it can be depicted by a signed graph whose positive edges indicate liking and the negative edges denote disliking. In a sigraph G, a path of length p between two vertices u and v is a sequence of edges P = a 0 a 1 ,a 1 a 2 ,...,a p−1 a p where u = a 0 and a p = v. Then, the sign of the path P is defined as posi- tive if the number of negative edges in P is even, and negative if the number of negative edges in P is odd. Precisely, s(P )=Π p−1 i=0 s(a i a i+1 ). Cycles represent a particular class of paths in which the end-vertices coincide. Therefore, cycles also have a corresponding sign. We say that a sigraph is balanced if and only if, all of its cycles are pos- itive. Equivalently, Harary proved in [3] that a sigraph is balanced if and only if all paths be- tween any two different vertices have the same sign. In [1] and [2] the authors defined different op- erations on sigraphs, based on the line graph of an unsigned graph, and it was analyzed under what circumstances do those operations derive into balanced sigraphs. We propose an opera- tion, also based on the notion of line graph of an unsigned graph, that always leads to a balanced sigraph. For the completeness of this introduction, we recall the definition of line graph in the case of (unsigned) graphs. Let G ′ =(V ′ ,E ′ ) be a graph, its line graph is denoted by L(G ′ ), where the The International Symposium on Parallel Architectures, Algorithms, and Networks 978-0-7695-3125-0/08 $25.00 © 2008 IEEE DOI 10.1109/I-SPAN.2008.15 141