A Note on Laplacian Coefficients of the Characteristic Polynomial of a Complete Graph

We have discovered the distinctive polynomial of the complete graph’s Laplacian matrix with this article in mind. The trace of the Laplacian matrix and the total number of vertices in the complete graph were used to determine the coefficients of the characteristic polynomial. Additionally, we have demonstrated that Laplacian eigenvalues can be used to get coefficients for identical characteristic polynomials. Can be used to get coefficients for identical characteristic polynomials.