On limit points of Laplacian spectral radii of graphs

Abstract The study of limit points of eigenvalues of adjacency matrices of graphs was initiated by Hoffman [A.J. Hoffman, On limit points of spectral radii of non-negative symmetric integral matrices, in: Y. Alavi et al. (Eds.), Lecture Notes Math., vol. 303, Springer-Verlag, Berlin, Heidelberg, New York, 1972, pp. 165–172]. There he described all of the limit points of the largest eigenvalue of adjacency matrices of graphs that are no more than 2 + 5 . In this paper, we investigate limit points of Laplacian spectral radii of graphs. The result is obtained: Let ω = 1 3 19 + 3 33 3 + 19 - 3 33 3 + 1 , β 0 = 1 and β n ( n ⩾ 1 ) be the largest positive root of P n ( x ) = x n + 1 - ( 1 + x + ⋯ + x n - 1 ) x + 1 2 . Let α n = 2 + β n 1 2 + β n - 1 2 . Then 4 = α 0 α 1 α 2 ⋯ are all of the limit points of Laplacian spectral radii of graphs smaller than lim n → ∞ α n = 2 + ω + ω - 1 ( = 4.38 + ) .