This paper discusses the properties of the characteristic polynomial of the complete graphs Kn, n=1, 2… respective to the adjacency matrices. Two different types of matrices, the adjacency matrix and the signless Laplacian matrix, are presented. A recurrence relation for computing the characteristic polynomials depending on the adjacency matrix is introduced. We deduce that the coefficients of the polynomials based on the two different matrices have a relationship with Pascal triangle. The coefficients are computed using Matlab program. Many other properties of these coefficients are discussed also.
Keywords: Laplacian matrix, Pascal triangle, Complete graph.