By Ali Reza Ashrafi, Mircea V. Diudea

This contributed quantity is electrified via the seminal discovery and identity of C60. beginning with a accomplished dialogue that includes graphene dependent nanostructures, next chapters contain topological descriptions of matrices, polynomials and indices, and a longer research of the symmetry and topology of nanostructures. Carbon allotropes equivalent to diamond and its connection to higher-dimensional areas is explored in addition to very important mathematical and topological concerns. additional subject matters lined comprise spontaneous symmetry breaking in graphene, polyhedral carbon buildings, nanotube junction energetics, and cyclic polyines as relations of nanotubes and fullerenes. This e-book is geared toward researchers lively within the examine of carbon fabrics technological know-how and technology.

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**Example text**

2003). V. Diudea and B. Szeﬂer The number of structures investigated for transitivity can be reduced by using an orthogonal edge-cutting procedure (see below). 2 Omega Polynomial by Edge-Cutting Procedures In bipartite graphs, an orthogonal edge-cutting procedure (Diudea 2010a, b, c; Diudea et al. 2010a, b; Gutman and Klavžar 1995; Klavžar 2008a, b) can be used to generate the ops. In performing a cut, take a straight line segment, orthogonal to the edge e, and intersect e and all its parallel edges (in a polygonal plane graph).

Some counterexamples for a recent open question on this topic are also presented; see Koorepazan-Moftakhar and Ashraﬁ (2015). We end this chapter with the following open question. Open Question Find a general formula for the class functions of fullerenes. Acknowledgment The ﬁrst and second authors are partially supported by the University of Kashan under Grant No. 464092/2. 3 An Algebraic Modiﬁcation of Wiener and Hyper–Wiener Indices and Their. . 49 References Ashraﬁ AR, Koorepazan-Moftakhar F (2014) Fullerenes and capped nanotubes: applications and geometry.

GL(n,C) is called a representation of G, where GL(n,C) denotes the set of all n Â n invertible matrices over the complex ﬁeld C. The mapping χ: G ! C given by χ(g) ¼ Trf(g) is called a character afforded by f. Suppose χ and ψ are characters of G. The inner product hχ,ψi is deﬁned as ⟨χ, ψ⟩ ¼ 1 X χðgÞ ψðgÞ: g2G jGj The character χ is called irreducible, if hχ,χi ¼ 1. It is well known that the number of irreducible characters is the same as the number of conjugacy classes in G. One of the main applications of group theory in science is studying symmetry of molecules.