Some physical and chemical indices of clique-inserted-lattices

The operation of replacing every vertex of an $r$-regular lattice $H$ by a complete graph of order $r$ is called clique-inserting, and the resulting lattice is called the clique-inserted-lattice of $H$. For any given $r$-regular lattice, applying this operation iteratively, an infinite family of $r$-regular lattices is generated. Some interesting lattices including the 3-12-12 lattice can be constructed this way. In this paper, we reveal the relationship between the energy and resistance distance of an $r$-regular lattice and that of its clique-inserted-lattice. As an application, the asymptotic energy per vertex and average resistance distance of the 3-12-12 and 3-6-24 lattices are computed. We also give formulae expressing the numbers of spanning trees and dimers of the $k$-th iterated clique-inserted lattices in terms of that of the original lattice. Moreover, we show that new families of expander graphs can be constructed from the known ones by clique-inserting

Spectrum and genus of commuting graphs of some classes of finite rings

The commuting graph of a non-commutative ring $R$ with center $Z(R)$ is a simple undirected graph whose vertex set is $R\setminus Z(R)$ and two vertices $x, y$ are adjacent if and only if $xy = yx$. In this paper, we compute the spectrum and genus of commuting graphs of some classes of finite rings

Asymptotic Laplacian-Energy-Like Invariant of Lattices

Let $\mu_1\ge \mu_2\ge\cdots\ge\mu_n$ denote the Laplacian eigenvalues of $G$ with $n$ vertices. The Laplacian-energy-like invariant, denoted by $LEL(G)= \sum_{i=1}^{n-1}\sqrt{\mu_i}$, is a novel topological index. In this paper, we show that the Laplacian-energy-like per vertex of various lattices is independent of the toroidal, cylindrical, and free boundary conditions. Simultaneously, the explicit asymptotic values of the Laplacian-energy-like in these lattices are obtained. Moreover, our approach implies that in general the Laplacian-energy-like per vertex of other lattices is independent of the boundary conditions.Comment: 6 pages, 2 figure

Spectrum of commuting graphs of some classes of finite groups

In this paper, we initiate the study of spectrum of the commuting graphs of finite non-abelian groups. We first compute the spectrum of this graph for several classes of finite groups, in particular AC-groups. We show that the commuting graphs of finite non-abelian AC-groups are integral. We also show that the commuting graph of a finite non-abelian group $G$ is integral if $G$ is not isomorphic to the symmetric group of degree $4$ and the commuting graph of $G$ is planar. Further it is shown that the commuting graph of $G$ is integral if the commuting graph of $G$ is toroidal

Stationary real solutions of the nonlinear Schr\"odinger equation on a ring with a defect

We analyze the 1D cubic nonlinear stationary Schr\"odinger equation on a ring with a defect for both focusing and defocusing nonlinearity. All possible $\delta$ and $\delta'$ boundary conditions are considered at the defect, computing for each of them the real eigenfunctions, written as Jacobi elliptic functions, and eigenvalues for the ground state and first few excited energy levels. All six independent Jacobi elliptic functions are found to be solutions of some boundary condition. We also provide a way to map all eigenfunctions satisfying $\delta$/$\delta'$ conditions to any other general boundary condition or point-like potential.Comment: 7 pages, 7 figure

Macroscopic electromagnetic response of metamaterials with toroidal resonances

Toroidal dipole, first described by Ia. B. Zeldovich [Sov. Phys. JETP 33, 1184 (1957)], is a distinct electromagnetic excitation that differs both from the electric and the magnetic dipoles. It has a number of intriguing properties: static toroidal nuclear dipole is responsible for parity violation in atomic spectra; interactions between static toroidal dipole and oscillating magnetic dipole are claimed to violate Newton's Third Law while non-stationary charge-current configurations involving toroidal multipoles have been predicted to produce vector potential in the absence of electromagnetic fields. Existence of the toroidal response in metamaterials was recently demonstrated and is now a growing field of research. However, no direct analytical link has yet been established between the transmission and reflection of macroscopic electromagnetic media and toroidal dipole excitations. To address this essential gap in electromagnetic theory we have developed an analytical approach linking microscopic and macroscopic electromagnetic response of a metamaterial and showed, using a case study, the key role of the toroidal dipole in shaping the electromagnetic properties of the metamaterial

Various energies of some super integral groups

In this paper, we obtain energy, Laplacian energy and signless Laplacian energy of the commuting graphs of some families of finite non-abelian groups.Comment: arXiv admin note: text overlap with arXiv:1608.0276

Mahler Measure and the Vol-Det Conjecture

The Vol-Det Conjecture relates the volume and the determinant of a hyperbolic alternating link in $S^3$. We use exact computations of Mahler measures of two-variable polynomials to prove the Vol-Det Conjecture for many infinite families of alternating links. We conjecture a new lower bound for the Mahler measure of certain two-variable polynomials in terms of volumes of hyperbolic regular ideal bipyramids. Associating each polynomial to a toroidal link using the toroidal dimer model, we show that every polynomial which satisfies this conjecture with a strict inequality gives rise to many infinite families of alternating links satisfying the Vol-Det Conjecture. We prove this new conjecture for six toroidal links by rigorously computing the Mahler measures of their two-variable polynomials.Comment: 29 pages. V2: Minor changes, fixed typos, improved expositio

Quantum electron transport in toroidal carbon nanotubes with metallic leads

A recursive Green's function method is employed to calculate the density-of-states, transmission function, and current through a 150 layer (3,3) armchair nanotorus (1800 atoms) with laterally attached metallic leads as functions of relative lead angle and magnetic flux. Plateaus in the transmissivity through the torus occur over wide ranges of lead placement, accompanied by enhancements in the transmissivity through the torus as magnetic flux normal to the toroidal plane is varied.Comment: 15 pages, 11 .eps figures, 1 black and white figure, 10 color figures, uses revtex4; manuscript presented at conference NSTI07 Nanotech 2007, May 20-24, 2007, Santa Clara, C

Approval Voting in Product Societies

In approval voting, individuals vote for all platforms that they find acceptable. In this situation it is natural to ask: When is agreement possible? What conditions guarantee that some fraction of the voters agree on even a single platform? Berg et. al. found such conditions when voters are asked to make a decision on a single issue that can be represented on a linear spectrum. In particular, they showed that if two out of every three voters agree on a platform, there is a platform that is acceptable to a majority of the voters. Hardin developed an analogous result when the issue can be represented on a circular spectrum. We examine scenarios in which voters must make two decisions simultaneously. For example, if voters must decide on the day of the week to hold a meeting and the length of the meeting, then the space of possible options forms a cylindrical spectrum. Previous results do not apply to these multi-dimensional voting societies because a voter's preference on one issue often impacts their preference on another. We present a general lower bound on agreement in a two-dimensional voting society, and then examine specific results for societies whose spectra are cylinders and tori.Comment: 12 pages, 8 figures; to appear, Amer. Math. Monthl