General Randić indices for matching and $\cal{L}$-characteristics polynomial of Starlike trees

Here we study the normalized Laplacian characteristics polynomial (L-polynomial) for trees and specifically for starlike trees. We describe how the L-polynomial of a tree depends on some topological indices. For which, we also define the higher order general Randić indices for matching and which are different from higher order connectivity indices. Finally we provide the multiplicity of the eigenvalue 1 for starlike trees. AMS classification: 05C50; 05C05 Keywards: Tree; Starlike tree; Normalized Laplacian; Characteristics Polynomial; Randić index; Matching; Eigenvalue 1.