Abstract: Using the properties of determinant, firstly, we gave the normalized Laplacian and normalized signless Laplacian characteristic polynomials based on  the characteristic polynomial of the Randic matrix of a graph G. Secondly, we gave the characteristic polynomials of Randic matrix, normalized Laplacian matrix and normalized signless Laplacian matrix by means of the characteristic polynomials of the induced subgraphs. Finally, we gave the characteristic polynomial of the Middle graph of a graph G.

Key words: normalization, Laplacian characteritic polynomial,  , signless Laplacian characteristic polynomial,  , Randic matrix