A Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [dij], where for i≠j, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for distinct nuclei. The aim of this paper is to compute the automorphism group of the Euclide...

Michael A. Stroscio Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, IL 60607 , USA Abstract This note describes the solution of the Helmholtz equation inside a nanotorus with uniform Dirichlet boundary conditions. The eigenfunction symmetry is discussed and the lower-order eigenvalues and eigenfunctions are shown. The similarity with the case of a ...

the edge detour index polynomials were recently introduced for computing theedge detour indices. in this paper we nd relations among edge detour polynomials for the2-dimensional graph of tuc4c8(s) in a euclidean plane and tuc4c8(s) nanotorus.

Three thermodynamically stable nano-sized complexes containing 42 Ln3+ ions in a single molecule were synthesized via self-assembly process. The compounds have torus or wheel-like architectures and the general formula Ln42L14(OAc)82(OH)30 (Ln = Eu, Tb, Gd, L deprotonated ortho-vanillin, OAc acetate). Eu Tb represent first reported visibly luminescent lanthanide nanotoroids. crystal structures o...

Abstract: The Hosoya polynomial of a molecular graph G is defined as ∑ ⊆ = ) ( } , { ) , ( ) , ( G V v u v u d G H λ λ , where d(u,v) is the distance between vertices u and v. The first derivative of H(G,λ) at λ = 1 is equal to the Wiener index of G, defined as ∑ ⊆ = ) ( } , { ) , ( ) ( G V v u v u d G W . The second derivative of ) , ( 2 1 λ λ G H at λ = 1 is equal to the hyper-Wiener index, d...

in this paper, the weighted szeged indices of cartesian product and corona product of twoconnected graphs are obtained. using the results obtained here, the weighted szeged indices ofthe hypercube of dimension n, hamming graph, c4 nanotubes, nanotorus, grid, t− fold bristled,sunlet, fan, wheel, bottleneck graphs and some classes of bridge graphs are computed.

the center (periphery) of a graph is the set of vertices with minimum (maximum)eccentricity. in this paper, the structure of centers and peripheries of some classes ofcomposite graphs are determined. the relations between eccentricity, radius and diameterof such composite graphs are also investigated. as an application we determinethe center and periphery of some chemical graphs such as nanotor...

The Szeged index of a graph G is defined as S z(G) = ∑ uv = e ∈ E(G) nu(e)nv(e), where nu(e) is number of vertices of G whose distance to the vertex u is less than the distance to the vertex v in G. Similarly, the revised Szeged index of G is defined as S z∗(G) = ∑ uv = e ∈ E(G) ( nu(e) + nG(e) 2 ) ( nv(e) + nG(e) 2 ) , where nG(e) is the number of equidistant vertices of e in G. In this paper,...