By Ali Reza Ashrafi, Mircea V. Diudea

ISBN-10: 331931582X

ISBN-13: 9783319315829

ISBN-10: 3319315846

ISBN-13: 9783319315843

This contributed quantity is electrified by means of the seminal discovery and identity of C60. beginning with a accomplished dialogue that includes graphene established nanostructures, next chapters comprise topological descriptions of matrices, polynomials and indices, and a longer research of the symmetry and topology of nanostructures. Carbon allotropes similar to diamond and its connection to higher-dimensional areas is explored besides vital mathematical and topological concerns. extra themes coated comprise spontaneous symmetry breaking in graphene, polyhedral carbon constructions, nanotube junction energetics, and cyclic polyines as family of nanotubes and fullerenes. This publication is aimed toward researchers energetic within the examine of carbon fabrics technological know-how and technology.

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**Example text**

On the other hand, the Cluj polynomial shows a single term, as in bipartite tori. It is the only found non-bipartite graph which shows the equality ΠðGÞ ¼ PI e ðGÞ (usually holding in bipartite graphs). 4); all these tori are non-co-graphs and non-partial cubes. The third torus in Fig. 4 is a non-bipartite torus, covered by odd faces, in a ((5,7)3) tessellation. 17)) (Diudea 2010a, b, c). 5 Cubic Net and Corresponding Cage The cubic net in Fig. 4 left is bipartite and for sure partial cube and co-graph; it shows the triple equality CI ðGÞ ¼ ΠðGÞ ¼ PI e ðGÞ.

1) limited to a single face. , belong to ocs). By induction, any newly added edge of ops will be parallel to the previous one and also co-distant. d. Corollary In a co-graph, all the edges belonging to an ops are topologically parallel. Observe the relation co is a particular case of the edge equidistant eqd relation. 3)) (Diudea et al. 2008; Ashraﬁ et al. 2008a, b): e eqd f ðiiÞ , d ðu; xÞ ¼ dðu; yÞ ¼ dðv; xÞ ¼ dðv; yÞ ð2:3Þ Recall that a graph is planar if it allows an embedding into the plane such that no two edges cross.

A fullerene graph is a cubic, planar, and 3-connected whose faces are all pentagons or hexagons. A fullerene is a molecule, that is, its molecular graph is a fullerene graph. In 1991, Graovac and Pisanski (1991) proposed an algebraic modiﬁcation of the Wiener index of a graph by considering its automorphism group. In this chapter, an algebraic modiﬁcation for the hyper–Wiener index is presented. These quantities will be computed for some classes of fullerenes. F. R. ir O. Ori Actinium Chemical Research, Via Casilina 1626/A, 00133 Rome, Italy Laboratory of Computational and Structural Physical-Chemistry for Nanosciences and QSAR, Department of Biology-Chemistry, Faculty of Chemistry, Biology, Geography, West University of Timis¸oara, Pestalozzi Str.

### Distance, Symmetry, and Topology in Carbon Nanomaterials by Ali Reza Ashrafi, Mircea V. Diudea

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