Inertia and biclique decompositions of joins of graphs

Throughout the paper, G denotes a graph that has no loops, but that may have multiple edges. The multi-set of edges of G is denoted by E(G). A graph G is simple if it has no (loops or) multiple edges. A biclique of G is a simple complete bipartite subgraph of G. A biclique decomposition of G is a collection of bicliques of G, such that each edge of G is in precisely one of the bicliques in the collection. We let b(G) denote the minimum number of bicliques needed in a biclique decomposition of G. If A = A(G) is an adjacency matrix of G, then