The signless Laplacian spectral radius of some strongly connected digraphs (Q1799428)

A \(\infty\)-digraph is a digraph consisting of many directed cycles with one common vertex. If Q is the signless Laplacian matrix of a strongly connected digraph G, the spectral radius of Q is called the signless Laplacian spectral radius of G. A \(\theta\)-graph is a graph consisting of three paths having the same end vertices. Likewise, a generalized strongly connected \(\theta\)-digraph can be defined. The present paper is to determine the unique digraph which attains the maximum or minimum signless Laplacian spectral radius among all \(\infty\)-digraphs and \(\theta\)-digraphs. In addition, the paper also characterizes the extremal digraph which achieves the maximum signless Laplacian spectral radius among these types of digraphs.