Spectral radius and signless Laplacian spectral radius of strongly connected digraphs

Abstract

Let D be a strongly connected digraph and A(D) be the adjacency matrix of D. Let diag(D) be the diagonal matrix with outdegrees of the vertices of D and Q(D) = diag(D) + A(D) be the signless Laplacian matrix of D. The spectral radius of Q(D) is called the signless Laplacian spectral radius of D, denoted by q(D). In this paper, we give sharp bound on q(D) with outdegree sequence and compare the bound with some known bounds, establish some sharp upper or lower bound on q(D) with some given parameter such as clique number, girth or vertex connectivity, and characterize the corresponding extremal digraph or proposed open problem. In addition, we also determine the unique digraph which achieves the minimum (or maximum), the second minimum (or maximum), the third minimum, the fourth minimum spectral radius and signless Laplacian spectral radius among all strongly connected digraphs, and answer the open problem proposed by Lin-Shu [H.Q. Lin, J.L. Shu, A note on the spectral characterization of strongly connected bicyclic digraphs, Linear Algebra Appl. 436 (2012) 2524-2530].

An Extending Result on Spectral Radius of Bipartite Graphs

“…The idea of the proof in Lemma 2.2 is to apply Perron-Frobenius Theorem for spectral radius to matrices that are similar to the adjacency matrix of G by diagonal matrices with variables on diagonals. Results using this powerful method are also in [5,6,7,8,9,12,13,14].…”

On the $A_α$ spectral radius and $A_α$ energy of digraphs

“…The adjacent spectral radius and signless Laplacian spectral radius of digraphs had received a lot of attention from researchers. For some papers and related results in these directions see [2,4,7,10,11,19,20] and the references therein. However, there is no much known about the A α spectral radius of digraphs.…”

On the $A_α$ spectral radius of strongly connected digraphs

“…Let D(G) be the diagonal matrix with outdegrees of vertices of G. The sum of A(G) and D(G) is called the signless Laplacian matrix Q(G), which has been extensively studied since then. More detailed information about this research see [6,9,19,20], and their references. Nikiforov [16] proposed to study the convex linear combinations of the adjacency matrix and diagonal matrix of degrees of undirected graphs, which give a unified theory of adjacency spectral and signless Laplacian spectral theories.…”

“…In [10], Lin and Shu characterized the digraph which has the maximum A 0 spectral radius among all strongly connected digraphs with given dichromatic number. In [6], Hong and You determined the digraph which achieves the minimum (or maximum) A 1 2 spectral radius among all strongly connected digraphs with some given parameters such as clique number, girth or vertex connectivity. In [20], Xi and Wang determined the extremal digraph with the maximum A 1 2 spectral radius among all strongly connected digraphs with given dichromatic number.…”

“…If α = 0, Li and Zhou [7] proved that θ(0, 1, n−3), θ(1, 1, n−4) and θ(0, 2, n−4) are the unique digraphs which achieve the second, the third and the fourth minimum A 0 spectral radius among all strongly connected digraphs, respectively. If α = 1 2 , Hong and You [6] determined that θ(0, 1, n−3), θ(1, 1, n−4) and θ(0, 2, n−4) also attain the second, the third and the fourth minimum A 1 2 spectral radii among all strongly connected digraphs, respectively.…”