Dominant eigenvalue and universal winners of digraphs

Abstract

Let A be a nonnegative irreducible matrix of order n and Ai(t) be the matrix obtained by increasing its ith diagonal entry by a positive number t. An index i ∈ [n] = {1, . . . , n} is called a universal winner for A, if, letting ρ(·) denote the spectral radius, ρ(Ai(t)) ≥ ρ(Aj (t)) for j ∈ [n] and for t ∈ (0, ∞). Let G be a strongly connected digraph with vertex set [n]. Let WG be the set of all nonnegative matrices whose underlying graph structure is G. We say a vertex u structurally dominates another vertex v in G, if ρ(Au(t)) ≥ ρ(Av(t)) for all t ∈ (0, ∞) and for all A ∈ WG. We characterize the class of digraphs G that do not have a vertex that structurally dominates all other vertices. We say two vertices u and v structurally tie if they structurally dominate each other in G. We supply an equivalent graph theoretic condition for the structural tying of two vertices in G. Let S ⊆ [n] be nonempty. We characterize the class of strongly connected digraphs G with vertex set [n] such that S is the set of universal winners for each A ∈ WG. We also characterize the strongly connected digraphs whose vertex set can be partitioned into subsets P1, P2, . . . , Pk such that vertices inside a part Pi structurally tie with each other and vertices of Pi structurally dominate vertices of Pj strictly (without tying) for i < j