Aα-spectral radius and measures of graph irregularity. Issue 20 (19th December 2022) - Ji, Yuqing; Wang, Jianfeng; Brunetti, Maurizio; Bian, Ning
ABSTRACT: Let G be a graph with order n and size m . For each real number α ∈ [ 0, 1 ], the A α -matrix of a graph G is defined as A α ( G ) := α D ( G ) + ( 1 − α ) A ( G ), where D ( G ) = d i a g ( d 1, …, d n ) is the degree matrix of G, and A ( G ) is its adjacency matrix. The functions 1 s ( G ) = ∑ 1 ≤ i ≤ n d i − 2 m n a n d v a r ( G ) = 1 n ∑ 1 ≤ i ≤ n d i − 2 m n 2 have been already used elsewhere as measures of graph irregularity. In this paper, we find several bounds for the spectral radius λ ( G ) of the matrix A α ( G ) . Especially, when G is non-empty we prove for α ∈ [ 0, 1 ) that 2 s 2 ( G ) 2 n 2 n ( 2 α − α 2 ) Δ 2 + 2 ( 1 − α ) 2 m ≤ λ ( G ) − 2 m n ≤ 1 + ( 2 α − α 2 ) n Ω ( G ) 2 + ( 1 − α ) 2 s ( G ), where Ω ( G ) = Δ ( G ) − ⌈ 2 m / n ⌉ + 1 and Δ ( G ) is the maximum vertex degree of G . These inequalities extend to the A α -matrix a Nikiforov's result involving the spectral radius of the adjacency matrix. In addition, for fixed order and size, we determine the graphs minimizing the degree variance v a r ( G ) and compute the A α -spectrum of some blown-up graphs.