The (distance) signless Laplacian spectral radius of digraphs with given arc connectivity

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 ^⁎.