A tree whose complement is not eigensharp Original Research Article

The minimum number of complete bipartite subgraphs needed to partition the edges of a graph G is denoted by b(G). A known lower bound on b(G) states that b(G) is at least the maximum of the number of positive and negative eigenvalues of the adjacency matrix A of G; that is b(G)greater-or-equal, slantedmax{n+(A),n−(A)}. When equality is attained G is said to be eigensharp. Using known necessary conditions for equality, it is shown that there is a tree on 14 vertices whose complement is not eigensharp. It is also shown that G is the eigensharp when G is the complement of a forest where each component is a path.