caa a22 4500 605523029 CHVBK 20210128100747.0 cr unu---uuuuu 210128e20151101xx s 000 0 eng 10.1007/s10958-015-2602-3 doi (NATIONALLICENCE)springer-10.1007/s10958-015-2602-3 Saati H. Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan, Iran aut Topological Index of Some Carbon Nanotubes and the Symmetry Group for Nanotubes and Unit Cells in Solids [Elektronische Daten] [H. Saati] The Padmakar-Ivan (PI) index of a graph G is defined as PI G = ∑ e ∈ G n e u e G + n e v e G , $$ \mathrm{PI}(G)={\displaystyle \sum_{e\in G}\left[{n}_{eu}\left(e\left|G\right.\right)+{n}_{ev}\left(e\left|G\right.\right)\right]}, $$ where n eu (e|G) is the number of edges of G lying closer to u than to v, n ev (e|G) is the number of edges of G lying closer to v than to u, and summation goes over all edges of G. Let G be a connected graph, n eu (e|G) be the number of vertices of G lying closer to u, and n ev (e|G) be the number of vertices of G lying closer to v. Then the Szeged index of G is defined as the sum of n eu (e|G)n ev (e|G) over the edges of G. The PI index of G is the Szeged-like topological index defined as the sum of [n eu (e|G) + n ev (e|G)], where n eu (e|G) is the number of edges of G lying closer to u than to v, n ev (e|G) is the number of edges of G lying closer to v than to u, and summation goes over all edges of G. Different types of symmetry groups are commonly used in chemistry. Point groups are used for molecules, whereas, for solids, the 230 space groups are used. Neither of these types of symmetry groups are suitable for representing unit cells in solids, the symmetry of which is intermediate between that of point groups and space groups representing the symmetry of unit cells in an infinite lattice; the third type of symmetry group must be used. An algorithmic method of generating these symmetry groups is described. It could be demonstrated that these groups are appropriate by applying the conventional symmetry. This technique has been used in the case of the two-dimensional graphite lattice. Because the new method generates symmetry tables using only the topology of the system, the symmetry properties of graphs could also be readily derived. Finally, the relationship between these groups and the other two types of groups is established. Springer Science+Business Media New York, 2015 Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 211/1(2015-11-01), 58-126 1072-3374 211:1<58 2015 211 10958 https://doi.org/10.1007/s10958-015-2602-3 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10958-015-2602-3 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Saati H. Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan, Iran aut NATIONALLICENCE 773 0- Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 211/1(2015-11-01), 58-126 1072-3374 211:1<58 2015 211 10958