2021/09/28 by Shu-Guang Guo, Rong Zhang
Mathematics · Computer Science · #Graph theory and applications #Matrix Theory and Algorithms #Graph Labeling and Dimension Problems
paper · doi:10.1080/03081087.2021.1981811
Let G be a simple undirected graph. For real number α∈[0,1], Nikiforov defined the Aα-matrix of G as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G respectively. In this paper, we obtain a sharp upper bound on the largest eigenvalue ρα(G) of Aα(G) for α∈[1/2,1). Employing this upper bound, we prove that ‘For connected G1 and G2 with n vertices and m edges, if the maximum degree Δ(G1)≥2α(1−α)(2m−n+1)+2α and Δ(G1)>Δ(G2), then ρα(G1)>ρα(G2)’. Let λα(G) denote the least eigenvalue of Aα(G). For α∈(1/2,1), we prove that ‘For two connected G1 and G2, if the minimum degree δ(G1)≤11−α−2 and δ(G1)<δ(G2), then λα(G1)<λα(G2)’.