An algebraic study of non classical fullerenes

Analysis of the Graovac–Pisanski Index of Some Polyhedral Graphs Based on Their Symmetry Group

“…The Wiener and GP-indices of several infinite classes of fullerene and polyhedral graphs have been computed in [19,20,23] as well as [24][25][26]. The aim of this paper is to explore these quantities for some new classes of graphs.…”

“…In [17] it is proved that if T is a tree, then W(T) ≤ W(T) and in [18] the authors showed if G is either a connected bipartite graph or a connected graph of even order, then W(G) is an integer. This difference between the Wiener index and GP-index was first considered in [19], and in [20] this quantity was computed for some families of polyhedral graphs. Knor et al [21] considered the class of trees and they proved that this value is non-negative.…”

“…For more details about the GP number of nanostructures, linear polymers, some classes of fullerenes and fullerene-like molecules, see [8,[10][11][12][13][14]19,20,[22][23][24][25][26][27][28][29][30][31]. However, the following result is crucial in the whole of this paper.…”

The Graovac–Pisanski index of zig-zag tubulenes and the generalized cut method

“…It was also pointed out that electronic properties of carbon systems are deeply connected to the topology of their graphs (see book [5], pp. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].…”

“…For a vertex-transitive graph the Graovac-Pisanski index coincides with the Wiener index. The previous work on the symmetries of different nanostructures can be found in [1,2,7,8] and particularly for the Graovac-Pisanski index in [4,14,15,20]. In addition, the cut method for this index was developed in [9], where it was proved that the computation of the Graovac-Pisanski index can be reduced to the computation of the Wiener indices of the appropriately weighted quotient graphs.…”