Strongly subspectral pairs in C50+10n and C60+12n fullerenes via a common generic graph

Abstract

A convenient scheme has been developed for drawing the Schlegel digrams (graphs) of C_50+10n and C_60+12n (n=integer) fullerenes maintaining five-fold and six-fold rotational symmetry, respectively. It has been found that after symmetry-factorisation, 10+2n eigenvalues common to C_50+10n and C_60+12n fullerenes can be obtained from a generic graph and for this reason (C₆₀, C₇₂), (C₇₀, C₈₄), (C₈₀, C₉₆) ... , corresponding to n=1, 2, 3, ... respectively, are strongly subspectral pairs having 12, 14, 16, ... common eigenvalues respectively. Of the 10+2n common eigenvalues, n+3 can be expressed in the analytic form: λ_j= -1+2 cos[jπ/(n+4)], j=1, 2, ... n+3. The other n+7 common eigenvalues can be obtained from a weighted linear chain whose characteristic polynomial can be easily derived by means of a recently developed scheme; a sample calculation for n=1 has been shown. Results for a number of subspectral pairs have been tabulated.