Additive compound graphs

Based on matrix theory notions, we assign to an undirected finite (in general, signed) graph G on n vertices and each integer k, 1 ⩽ k ⩽ n, the kth additive compound graph G[k]. This is again an undirected signed graph on ( n k ) vertices. We investigate the basic properties of these graphs, e.g. show that they preserve connectedness of G, prove that the path Pn is the only connected graph G with all edges positive for which G[2] has only positive edges. The corresponding graphs Pn[k] as well as their spectral properties are completely described.