Mathematical fullerenes are convex polyhedra with pentagonal and hexagonal faces only. By Euler-Eberhard theorem, any fullerene with n vertices has exactly 12 pentagonal and n/2?10 hexagonal facets. The number of combinatorially non-equivalent fullerenes (isomers) with n vertices (playing the role of carbon atoms in chemistry) grows as O(n9), making statistical approaches essential for understanding their properties. This talk presents a study of spectral properties of fullerenes and related carbon nanos- tructures (graphene and nanotubes). We introduce random eigenvalues ?(Tn) of dual fullerene graphs obtained by choosing an eigenvalue uniformly from the spectrum of its adjacency matrix. We derive explicit (asymptotic as n ? ?) probability distribution functions and moment generating functions for these random eigenvalues, revealing intricate dependence on the combinatorial structure of the hexagonal planar lattice (graphene). For carbon nanotubes, classified by their chiral indices (p, q) into armchair (p = q), zigzag (q = 0), and chiral (p > q > 0) types, we establish exact formulas for the probability density functions of random eigenvalues in the infinite nanotube limit. The analysis distinguishes between metallic and semiconducting nanotubes based on the divisibility condition of p ? q by 3. We prove weak convergence of random eigenvalues of infinite (p, q)-nanotubes to the limiting distribution of random eigenvalues of infinite graphene as p + q ? ?. Finally, if time permits, we present some open problems. The results combine methods from spectral graph theory, probability theory, and combinatorics, offering new perspectives on the mathematical structure of carbon nanomaterials.