Spectral characterization conjecture for fullerene isomers

Let $C_n$-isomers be fullerene isomers with dual graphs $T_n$. Write $T_n^5$ and $T_n^6$ for the subgraphs induced by the pentagons and hexagons, respectively, and let $\sigma(G)$ denote the spectrum of a graph $G$. A value of $n$ is feasible when fullerene isomers with that number of vertices exist.

Spectral characterization conjecture. (a) For all feasible $n\ne44$, two $C_n$-isomers are isomorphic if and only if they are cospectral with respect to $T_n$.

(b) For all feasible $n\geq54$, two $C_n$-isomers are isomorphic if and only if they are cospectral with respect to $T_n^6$.

(c) For any feasible $n$, at least one of the spectra $\sigma(T_n)$, $\sigma(T_n^5)$, and $\sigma(T_n^6)$ is unique for all $C_n$-isomers.

These claims summarize empirical findings about the uniqueness of spectra for fullerene isomers and propose spectral criteria for distinguishing isomers, including criteria based on the pentagon and hexagon subgraphs. Their general validity is left unresolved by the reported computations.