Merging the A- and Q-spectral theories for digraphs

Abstract

Let G be a digraph and A(G) be the adjacency matrix of G. Let D(G) be the diagonal matrix with outdegrees of vertices of G. For any real α ∈The largest modulus of the eigenvalues of A α (G) is called the A α spectral radius of G. In this paper, we determine the digraphs which attain the maximum (or minimum) A α spectral radius among all strongly connected digraphs with given parameters such as girth, clique number, vertex connectivity or arc connectivity. We also discuss a number of open problems.

Trees with Given Independence Number Maximizing the A -Spectral Radius

“… is an eigenvalue of () AB  with multiplicity 1 nd , for a bug B of order n and diameter d ; Lin et al [15] characterized the extremal graphs with maximal A spectral radius for fixed order and cut vertices (or fixed order and matching number); Nikiforov et al [16] studied the distribution of the entries of Perron vectors along pendent paths in graphs for A  -spectral radius; Li et al [17] characterized all extremal trees and extremal unicyclic graphs with the maximum A  -spectral radius for prescribed degree sequence, respectively. For further results, one can refer to [18][19][20].…”

Some $α$-spectral extremal results for some digraphs

“…minimal) α-spectral radius among all strongly connected bicyclic digraphs. In [12], Xi and So determined the digraphs which attains the maximal (or minimal) α-spectral radius among all strongly connected digraphs with given parameters such as girth, clique number, vertex connectivity or arc connectivity. In [13], Hong and You determined the unique digraph with the minimal (or maximal), the second minimal (or maximal), the third minimal, the fourth minimal adjacency spectral radius and signless Laplacian spectral radius among all strongly connected digraphs.…”