Spectral Polynomials and Spectra of Graphs Beyond Cubic and Icosahedral Symmetries: n-Octahedra, n-Cubes, Symmetric and Semi-Symmetric Graphs, Giant Fullerene Cages and Generalized Petersen Graphs

We report the results of our computations of the spectral polynomials and spectra of a number of graphs possessing automorphism symmetries beyond cubic and icosahedral symmetries. The spectral (characteristic) polynomials are computed in fully expanded forms. The coefficients of these polynomials contain a wealth of combinatorial information that finds a number of applications in many areas including nanomaterials, genetic networks, dynamic stereochemistry, chirality, and so forth. This study focuses on a number of symmetric and semi-symmetric graphs with automorphism groups of high order. In particular, Heawood, Coxeter, Pappus, Möbius–Kantor, Tutte–Coxeter, Desargues, Meringer, Dyck, n-octahedra, n-cubes, icosahedral fullerenes such as C80(Ih), golden supergiant C240(Ih), Archimedean (Ih), and generalized Petersen graphs up to 720 vertices, among others, have been studied. The spectral polynomials are computed in fully expanded forms as opposed to factored forms. Several applications of these polynomials are briefly discussed.