Spectral radius of strongly connected digraphs

Abstract

We determine the digraphs which achieve the second, the third and the fourth minimum spectral radii respectively among strongly connected digraphs of order n ≥ 4, and thus we answer affirmatively the problem whether the unique digraph which achieves the minimum spectral radius among all strongly connected bicyclic digraphs of order n achieves the second minimum spectral radius among all strongly connected digraphs of order n for n ≥ 4 proposed in [H. Lin, J. Shu, A note on the spectral characterization of strongly connected bicyclic digraphs, Linear Algebra Appl. 436 (2012) 2524-2530. We also discuss the strongly connected bicyclic digraphs with small and large spectral radii respectively.

On the $A_α$ spectral radius of strongly connected digraphs

“…In Section 4, we determine the extremal digraph which attains the minimum A α spectral radius of strongly connected bipartite digraphs which contain a complete bipartite subdigraph. The results in our paper generalize some results in [2,4,7,9,14].…”

“…The most studied problems are those of characterization of extremal graphs, such as determine the maximum or minimum spectral (signless Laplacian spectral) radius over various families of graphs. Recently, in [12], Lin et al determined the digraphs with the minimum A 0 spectral radius among all strongly connected digraphs with given clique number and girth. In [8], Lin and Drury gave the extremal digraphs with the maximum A 0 radius among all strongly connected digraphs with given arc connectivity.…”

Spectral radius and signless Laplacian spectral radius of strongly connected digraphs

“…Note that ←→ K n is the unique digraph which achieves the maximum spectral radius n − 1 among all strongly connected digraphs, we have the following result by Theorem 4.2 in [15] and Lemma 2.3.…”

“…In Section 3 ∼ Section 6, we use some similar technique which applied in [15]. Although there are some defects in Section 2 ∼ Section 3 in [15] which can be revised by similar proofs of this paper, the results are well done and there are some useful techniques which can be used to study strongly connected digraphs. Now in this section, we only show that there are more results can be obtained on the spectral radius of strongly connected digraphs.…”

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.…”