Spectral problem of fullerene vibrations

Abstract

Small vibrations of a graph of fullerene (truncated icosahedron) is considered each edge of which is a so-called Stieltjes string (a massless thread bearing finite number of point masses) symmetric with respect to its midpoint. The spectral problem is obtained by imposing the continuity and balance of forces conditions at the vertices. It is shown that when all the edges of the graph are the same then due to the symmetry of the problem there are multiple eigenvalues. The maximal multiplicity of an eigenvalue of such problem is 32, exactly the value which is maximal for cyclically connected graphs, i.e. 𝜇 + 1 where 𝜇 is the cyclomatic number of the graph.