On Products and Line Graphs of Signed Graphs, their Eigenvalues and Energy

In this article we examine the adjacency and Laplacian matrices and their eigenvalues and energies of the general product (non-complete extended $p$-sum, or NEPS) of signed graphs. We express the adjacency matrix of the product in terms of the Kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor signed graphs. For the Cartesian product we characterize balance and compute expressions for the Laplacian eigenvalues and Laplacian energy. We give exact results for those signed planar, cylindrical and toroidal grids which are Cartesian products of signed paths and cycles. We also treat the eigenvalues and energy of the line graphs of signed graphs, and the Laplacian eigenvalues and Laplacian energy in the regular case, with application to the line graphs of signed grids that are Cartesian products and to the line graphs of all-positive and all-negative complete graphs.Comment: 30 page

Applications of Structural Balance in Signed Social Networks

We present measures, models and link prediction algorithms based on the structural balance in signed social networks. Certain social networks contain, in addition to the usual 'friend' links, 'enemy' links. These networks are called signed social networks. A classical and major concept for signed social networks is that of structural balance, i.e., the tendency of triangles to be 'balanced' towards including an even number of negative edges, such as friend-friend-friend and friend-enemy-enemy triangles. In this article, we introduce several new signed network analysis methods that exploit structural balance for measuring partial balance, for finding communities of people based on balance, for drawing signed social networks, and for solving the problem of link prediction. Notably, the introduced methods are based on the signed graph Laplacian and on the concept of signed resistance distances. We evaluate our methods on a collection of four signed social network datasets.Comment: 37 page

Integer symmetric matrices having all their eigenvalues in the interval [-2,2]

We completely describe all integer symmetric matrices that have all their eigenvalues in the interval [-2,2]. Along the way we classify all signed graphs, and then all charged signed graphs, having all their eigenvalues in this same interval. We then classify subsets of the above for which the integer symmetric matrices, signed graphs and charged signed graphs have all their eigenvalues in the open interval (-2,2).Comment: 33 pages, 18 figure