Bounds on the spectral radii of digraphs in terms of walks

abstractLet G=(V,E)G=(V,E) be a digraph with nn vertices and mm arcs without loops and multiarcs, V={v1,v2,…,vn}V={v1,v2,…,vn}. Denote by ρ(G)ρ(G) the largest eigenvalue of its adjacency matrix, Wk(i)Wk(i) the number of kk-walks from vertex vivi. In this paper, we prove that the inequalities minWp+q(i)Wq(i):vi∈V⩽ρp(G)⩽maxWp+q(i)Wq(i):vi∈V hold for every integers p⩾1,q⩾0. If GG is strongly connected, then each equality holds iff Wp+q(1)Wq(1)=Wp+q(2)Wq(2)=⋯=Wp+q(n)Wq(n) . Furthermore, we have limp→∞Wp+q(i)Wq(i)1p=ρ(G) for every integers p⩾1,q⩾0 and each vertex vi∈V(G)vi∈V(G).