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.