Inertia and biclique decompositions of joins of graphs

We characterize the inertia of A+B for Hermitian matrices A and B when the rank of B is one. We use this to characterize the inertia of a partial join of two graphs. We then provide graph joins G for which the minimum number of complete bipartite graphs needed in a partition of the edge multi-set of G is equal to the maximum of the number of positive and negative eigenvalues of G.