On the A_α spectral radius and A_α energy of digraphs

2021/07/14 by Xi, Weige
#05C50 #15A18 #Combinatorics (math.CO) #FOS: Mathematics #G.2.2

paper · doi:10.48550/arxiv.2107.06470

Abstract

Let G be a digraph with adjacency matrix A(G) and outdegrees diagonal matrix D(G). For any real α∈[0,1], the A_α matrix A_α(G) of a digraph G is defined as A_α(G)=αD(G)+(1-α)A(G). The eigenvalue of A_α(G) with the largest modulus is called the A_α spectral radius of G. In this paper, we first give some upper bounds for the A_α spectral radius of a digraph and we also characterize the extremal digraphs attaining these bounds. Moreover, we define the A_α energy of a digraph G as E^A_α(G)=∑_i=1ⁿ(λ^α_i(G))², where n is the number of vertices and λ^α_i(G) (i=1,2,…,n) are the eigenvalues of A_α(G). We obtain a formula for E^A_α(G), and give a lower and upper bounds for E^A_α(G) and characterize the extremal digraphs that attain the lower and upper bounds. Finally, we characterize the extremal digraphs with maximum and minimum A_α energy among all directed trees and unicyclic digraphs, respectively.

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