On the structure of Laplacian characteristic polynomial of circulant graphs

The present work deals with the characteristic polynomial of Laplacian matrix for circulant graphs. We show that it can be decomposed into a finite product of algebraic function evaluated at the roots of a linear combination of Chebyshev polynomials. As an important consequence of this result, we get the periodicity of characteristic polynomials evaluated at the prescribed integer values. Moreover, we can show that the characteristic polynomials of circulant graphs are always perfect squares up to explicitly given linear factors.