On the spectra of cycles and paths

Abstract

Energy of a graph was defined by E. Hückel as the sum of absolute values of the eigenvalues of the adjacency matrix during the search for a method to obtain approximate solutions of Schr¨odinger equation which include the energy of the corresponding system for a class of molecules. The set of eigenvalues is called the spectrum of the graph and the spectral graph theory dealing with spectrums is one of the most interesting subareas of graph theory. There are a lot of results on the energy of many graph types. Two classes, cycles and paths, show serious differences from others as the eigenvalues are trigonometric algebraic numbers. Here, we obtain the polynomials and recurrence relations for the spectral polynomials of these two graph classes. In particular, we prove that one can obtain the spectra of C2n and P2n+1 without detailed calculations just in terms of the spectra of Cn and Pn, respectively.