In graph theory, lattices are used when some structural part of the repeats itself finitely or infinitely many times. They have applications in complex analysis and geometry mathematics, also natural chemical theory. As a lattice can be taken as graph, it is possible to use them study large networks. Recently, topological indices pentagonal chains studied here, we double similarly that work. We...

Large carbon molecules, discovered at the end of the last century, are called fullerenes. The most famous of these, C60 has the structure of the soccer ball: the seams represent chemical bonds and the points where three seams come together represent the carbon atoms. We use the term fullerene to represent all mathematically possible structures for the chemical fullerenes: trivalent plane graphs...

fullerenes are closed−cage carbon molecules formed by 12 pentagonal and n/2 – 10hexagonal faces, where n is the number of carbon atoms. patrick fowler in his lecture inmcc 2009 asked about the wiener index of fullerenes in general. in this paper werespond partially to this question for an infinite class of fullerenes with exactly 10ncarbon atoms. our method is general and can be applied to full...

We present an optimal single-winner preferential voting system, called the “GT method” because of its interesting use of symmetric two-person zero-sum game theory to determine the winner. Game theory is not used to describe voting as a multi-player game between voters, but rather to define when one voting system is better than another one. The cast ballots determine the payoff matrix, and optim...

Isolated, gigantic tin clusters of 56 atoms are discovered in the ternary compound Ba(16)Na(204)Sn(310) (cubic, F(-)43m, Z = 1, a = 25.2041(8) A) made by direct fusion of the elements at 800 degrees C. The cluster, made of four face-fused pentagonal dodecahedra, has 36 pentagonal faces and 90 edges, and resembles a concave fullerene "dented" at four places. It is made of three- and four-bonded ...

The cowpea chlorotic mottle virus (CCMV) has a protein cage, or capsid, which encloses its genetic material. The structure of the capsid consists of 180 copies of a single protein that self-assemble inside a cell to form a complete capsid with icosahedral symmetry. The icosahedral surface can be naturally divided into pentagonal and hexagonal faces, and the formation of either of these faces ha...

Despite relevance to disparate areas such as cloud microphysics and tribology, major gaps in the understanding of the structures and phase transitions of low-dimensional water ice remain. Here, we report a first principles study of confined 2D ice as a function of pressure. We find that at ambient pressure hexagonal and pentagonal monolayer structures are the two lowest enthalpy phases identifi...

We derive a formula for p(n) (the number of partitions n) in terms the partial Bell polynomials using Faà di Bruno’s and Euler’s pentagonal theorem.

We explore the structure of the maximum vertex independence sets in fullerenes: plane trivalent graphs with pentagonal and hexagonal faces. At the same time, we will consider benzenoids: plane graphs with hexagonal faces and one large outer face. In the case of fullerenes, a maximum vertex independence set may constructed as follows: (i) Pair up the pentagonal faces. (ii) Delete the edges of a ...

We study the S=1/2 Heisenberg antiferromagnet on floret pentagonal lattice by numerical diagonalization method. This system shows various behaviours that are different from of Cairo-pentagonal-lattice antiferromagnet. The ground-state energy without magnetic field and magnetization process this reported. Magnetization plateaux appear at one-ninth height saturation magnetization, one-third heigh...