Construction of planar graphs for IPR fullerenes using 5- and 6-fold rotational symmetry: some eigenspectral analysis

Construction and utilisation of planar graphs of two series of IPR fullerenes through the use of threefold rotational symmetry

“…It is further shown that this procedure can easily be used to obtain the symmetry point groups and the number of 13 C NMR signals of the fullerenes and the generic graphs that yield the Hü ckel molecular orbital (HMO) eigenvalues of the fullerenes corresponding to the first block of the Davidson-Shen algorithm. It may be noted that in the present C 60ϩ12n fullerene series, the point group changes with increased n in a manner that is different from the C 60ϩ12n fullerenes obtained in our earlier works [44,45].…”

“…Repeat (a), (b), (c), one by one, up to the [3(nϩ2)/2]th circle for even n or up to the [(3nϩ7)/2]th circle for odd n. Now place (i) three horizontal and three vertical K 2 at the points of intersection of R 1 with the [(3nϩ8)/2]th circle alternantly for even n, or of R 2 with the [(3nϩ9)/2]th circle for odd n; and (ii) six vertical K 2 's at the points of intersection of R 2 with the [(3nϩ10)/2]th circle alternantly for even n, or at the points of intersection of R 1 with the [(3nϩ11)/2]th circle alternantly for odd n. Finally, the desired planar graphs of the IPR fullerenes of general formula C 60ϩ12n are obtained by joining the free ends of the K 2 to the nearest vertices to produce only pentagonal or hexagonal rings as shown in Figure 3 for C 108 whose point group is D 3h (in contrast to D 6 h of the isomeric C 108 obtained in our previous work [45]). …”

“…These are the graphs whose eigenvalues are contained in the first block obtained by subjecting the whole fullerene graph to the Davidson-Shen [40,41] algorithm, a compact graphical procedure for which was developed and illustrated in Ref. [45]. With the same procedure, the generic graphs of C 60ϩ12n and C 60ϩ18n fullerenes have been constructed in the present work by taking out the cut portion [45] generated by the threefold rotational symmetry axis passing through the center of the planar graphs drawn.…”

“…In our earlier works [44,45], construction of the generic graphs of C 50ϩ10n and C 60ϩ12n fullerenes with five-and sixfold rotational symmetry were discussed in detail. These are the graphs whose eigenvalues are contained in the first block obtained by subjecting the whole fullerene graph to the Davidson-Shen [40,41] algorithm, a compact graphical procedure for which was developed and illustrated in Ref.…”

“…A new algorithm for drawing graphs of IPR fullerenes by maintaining five-and sixfold rotational symmetry was developed by our group [44]; strong subspectrality among two series of fullerenes (C 50ϩ10n and C 60ϩ12n ) was shown using the Davidson-Shen algorithm. Recently [45] we developed a more systematic algorithm for construction of IPR fullerenes by using five-and sixfold rotational symmetry, such that point group oscillations, the number of 13 C nuclear magnetic resonance (NMR) signals, and the generic graphs for obtaining a large portion of the graph eigenspectra are revealed more easily. The present study extends this algorithm to threefold rotational symmetry, in the process, we find two new series of IPR fullerenes (C 60ϩ12n and C 60ϩ18n ).…”