Abstract
We describe the structure of the characteristic polynomial \(\chi _{\mathscr L}\) of the Laplacian matrix for a circulant graph withnon-fixed jumps. We represent the characteristic polynomial in the form of the product ofalgebraic functions involving roots of linear combinations of Chebyshev polynomials of the firstkind. We show that \(\chi _{\mathscr L}\) isthe product of the square of a polynomial with integer coefficients and explicitly described linearpolynomials with integer coefficients. We suggest a formula for the number of rooted spanningforests in such a graph.
