2022/10/29 by Liu, Xichan, Wang, Ligong
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2210.16480
Let G be a graph with adjacency matrix A(G) and degree diagonal matrix D (G). In 2017, Nikiforov defined the matrix Aα(G) = αD(G) + (1-α)A(G) for any real α∈[0,1]. The largest eigenvalue of Aα(G) is called the Aα spectral radius or the Aα-index of G. Let Gn,kd be the set of k-connected graphs of order n with diameter d. In this paper, we determine the graphs with maximum Aα spectral radius among all graphs in Gn,kd for any α∈[0,1), where k≥2 and d≥2. We generalizes the results about adjacency matrix of Theorem 3.6 in [P. Huang, W.C. Shiu, P.K. Sun, Linear Algebra Appl., 488 (2016) 350--362] and the results about signless Laplacian matrix of Theorem 3.4 in [P. Huang, J.X. Li, W.C. Shiu, Linear Algebra Appl., 617 (2021) 78--99]. Furthermore, we also obtain the upper and lower bounds of the extremal graph in Gn,kd.