LAUSR

Abstract Let G ‾ n , k denote the set of strongly connected digraphs with order n and arc connectivity k, and let G ‾ n , k ⁎ denote the set of digraphs in G ‾ n , k with all vertices having outdegree and indegree greater than k. In this paper, we determine the unique digraph with the maximum signless Laplacian spectral radius among all digraphs in G ‾ n , k . We also determine the unique one with the maximum signless Laplacian spectral radius among all digraphs in G ‾ n , k ⁎ with k = 1 , 2 . For the general case, we propose a conjecture on the maximum signless Laplacian spectral radius among all digraphs in G ‾ n , k ⁎ . Moreover, we characterize the extremal digraph achieving the minimum distance signless Laplacian spectral radius among all digraphs in G ‾ n , k . We also characterize the extremal digraph achieving the minimum distance signless Laplacian spectral radius among all digraphs in G ‾ n , k ⁎ with k = 1 , 2 . For the general case, we propose a conjecture on the minimum distance signless Laplacian spectral radius among all digraphs in G ‾ n , k ⁎ .