The A_α spectral radius with given independence number n-4

2022/10/29 by Xichan Liu, Liu, Xichan, Ligong Wang +1 · 1 citation
Materials Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Nonlinear Optical Materials Research #Phase-change materials and chalcogenides

paper · pdf · doi:10.48550/arxiv.2210.16466

openalex publication_date 2022/10/29 · openalex created_date 2022/11/06 · openalex updated_date 2026/07/28

Abstract

Let G be a graph with adjacency matrix A(G) and degree diagonal matrix D (G). In 2017, Nikiforov [Appl. Anal. Discrete Math., 11 (2017) 81--107] defined the matrix A_α(G) = αD(G) + (1-α)A(G) for any real α∈[0,1]. The largest eigenvalue of A(G) is called the spectral radius of G, while the largest eigenvalue of A_α(G) is called the A_α spectral radius of G. Let G_n,i be the set of graphs of order n with independence number i. Recently, for all graphs in G_n,i having the minimum or the maximum A, Q and A_α spectral radius where i∈\1,2,\lfloor(n)/(2)\rfloor \lceil(n)/(2)\rceil+1,n-3,n-2,n-1\, there are some results have been given by Xu, Li and Sun et al., respectively. In 2021, Luo and Guo [Discrete Math., 345 (2022) 112778] determined all graphs in G_n,n-4 having the minimum spectral radius. In this paper, we characterize the graphs in G_n,n-4 having the minimum and the maximum A_α spectral radius for α∈[(1)/(2),1), respectively.

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