On the minimum rank of the join of graphs and decomposable graphs

For a given undirected graph G, the minimum rank of G is defined to be the smallest possible rank over all real symmetric matrices A whose (i, j)th entry is nonzero whenever i ≠ j and {i, j} is an edge in G. In this work we consider joins and unions of graphs, and characterize the minimum rank of such graphs in the case of ‘balanced inertia’. Several consequences are provided for decomposable graphs, also known as cographs.