The Signless Laplacian Spectral Radius of Some Strongly Connected Digraphs

Let \(\vec G\) be a strongly connected digraph and Q( \(\vec G\)) be the signless Laplacian matrix of \(\vec G\). The spectral radius of Q(\(\vec G\)) is called the signless Lapliacian spectral radius of \(\vec G\). Let \({\tilde \infty _1}\)-digraph and \({\tilde \infty _2}\)-digraph be two kinds of generalized strongly connected 1-digraphs and let \({\tilde \theta _1}\)-digraph and \({\tilde \theta _2}\)-digraph be two kinds of generalized strongly connected µ-digraphs. In this paper, we determine the unique digraph which attains the maximum(or minimum) signless Laplacian spectral radius among all \({\tilde \infty _1}\)-digraphs and \({\tilde \theta _1}\)-digraphs. Furthermore, we characterize the extremal digraph which achieves the maximum signless Laplacian spectral radius among \({\tilde \infty _2}\)-digraphs and \({\tilde \theta _2}\)-digraphs, respectively.