Spectral Properties of Reducible Nonnegative and Eventually Nonnegative Matrices

Let η = (η1, η2, . . . , ηt) and ν = (ν1, ν2, . . . , νt) be two sequences of nonnegative integers. (Append zeros if necessary to the end of the shorter sequence so that they are the same length.) We say that ν is majorized by η if ∑j l=1 νl ≤ ∑j l=1 ηl for all 1 ≤ j ≤ t and tl=1 νl = ∑t l=1 ηl. We write ν 1 η. Let Γ = (V,E) be a graph where V is a finite vertex set and E ⊆ V ×V is an edge set. A path from j to m is a sequence of vertices j = v1, v2, ..., vt = m with (vl, vl+1) ∈ E for l = 1, ..., t − 1. A simple path is a path where the vertices are pairwise distinct. The empty path will be considered to be a simple path linking every vertex to itself. The path v1, v2, ..., vt is a cycle if v1 = vt and v1, v2, . . . , vt−1 is a simple path.