Some $α$-spectral extremal results for some digraphs

Abstract

In this paper, we characterize the extremal digraphs with the maximal or minimal α-spectral radius among some digraph classes such as rose digraphs, generalized theta digraphs and tri-ring digraphs with given size m. These digraph classes are denoted by R k m , Θ k (m) and (m) respectively. The main results about spectral extremal digraph by Guo and Liu in [1] and Li and Wang in [2] are generalized to α-spectral graph theory. As a by-product of our main results, an open problem in [2] is answered. Furthermore, we determine the digraphs with the first three minimal αspectral radius among all strongly connected digraphs. Meanwhile, we determine the unique digraph with the fourth minimal α-spectral radius among all strongly connected digraphs for 0 ≤ α ≤ 1 2 .

On the spectral radius of unicyclic and bicyclic graphs with a fixed diameter

“…Lemma 2.2. [14] Let A and B be two nonnegative matrices of order n with A ≥ B and A = B. Then the following holds:…”