A bound for the A_α-spectral radius of a connected graph after vertex deletion

School of Mathematics and Statistics, Central China Normal University, Wuhan, P.R. China

^* Corresponding author: she_tao@163.com

Received: 14 March 2022
Accepted: 4 October 2022

Abstract

G is a simple connected graph with adjacency matrix A(G) and degree diagonal matrix D(G). The signless Laplacian matrix of G is defined as Q(G) = D(G) + A(G). In 2017, Nikiforov [1] defined the matrix A_α(G) = α D(G) + (1 − α)A(G) for α ∈ [0,1]. The A_α-spectral radius of G is the maximum eigenvalue of A_α (G). In 2019, Liu et al. [2] defined the matrix Θ_k(G) as Θ_k (G) = kD(G) + A(G), for k ∈ ℝ. In this paper, we present a new type of lower bound for the A_α-spectral radius of a graph after vertex deletion. Furthermore, we deduce some corollaries on Θ_k (G), A(G), Q(G) matrices.