Graph theoretic aspects of maximizing the spectral radius of nonnegative matrices Original Research Article

Let ρi(t) = ρ(A + tEii) denote the spectral radius of the sum of an irreducible nonnegative matrix A and a matrix tEii that has a single nonzero entry, namely t > 0 in the i, i position. We consider qualitative aspects of maximizing ρi(t), especially identifying maximizing indices i, and indices i and j that tie [i.e., ρi(t) = ρj(t) for all t > 0]. If the digraph of A is a directed cycle, then all vertices tie; whereas if the digraph of A is a star, then the center is the unique maximizing vertex. When A is the (0, 1) adjacency matrix of a graph, we give sufficient conditions in terms of the orbits of vertices for a tie. For complete bipartite graphs and for paths, vertices i are identified that maximize ρi(t) for all t > 0. However, even for a tree, it is not in general true that some fixed vertex i maximizes ρi(t) for all t > 0.