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

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⁎.