On limit points of Laplacian spectral radii of graphs

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=1319+3333+19-3333+1, β0=1β0=1 and βn(n⩾1) be the largest positive root ofPn(x)=xn+1-(1+x+⋯+xn-1)x+12.Let αn=2+βn12+βn-12. Then4=α0<α1<α2<⋯4=α0<α1<α2<⋯are all of the limit points of Laplacian spectral radii of graphs smaller than limn→∞αn=2+ω+ω-1(=4.38+)limn→∞αn=2+ω+ω-1(=4.38+).