The ab-method, spectra, and spanning trees of some Toeplitz graphs

This paper introduces the ab-method, a parametrized determinant technique for studying spectra and spanning trees of graphs. The method encodes several graph matrices in a single determinant framework by assigning parameters to the off-diagonal entries and choosing the diagonal entries according to the problem under consideration. The method is applied to several classes of Toeplitz graphs, and closed-form adjacency and Laplacian spectra are derived for the corresponding infinite families. It is also shown that suitable Laplacian cofactors, obtained by deleting the last row and the last column, can be evaluated within the same framework. By the Matrix–Tree Theorem, this gives closed formulae for the number of spanning trees of the corresponding Toeplitz graphs.