956 research outputs found
Glassy correlations and microstructures in randomly crosslinked homopolymer blends
We consider a microscopic model of a polymer blend that is prone to phase
separation. Permanent crosslinks are introduced between randomly chosen pairs
of monomers, drawn from the Deam-Edwards distribution. Thereby, not only
density but also concentration fluctuations of the melt are quenched-in in the
gel state, which emerges upon sufficient crosslinking. We derive a Landau
expansion in terms of the order parameters for gelation and phase separation,
and analyze it on the mean-field level, including Gaussian fluctuations. The
mixed gel is characterized by thermal as well as time-persistent (glassy)
concentration fluctuations. Whereas the former are independent of the
preparation state, the latter reflect the concentration fluctuations at the
instant of crosslinking, provided the mesh size is smaller than the correlation
length of phase separation. The mixed gel becomes unstable to microphase
separation upon lowering the temperature in the gel phase. Whereas the length
scale of microphase separation is given by the mesh size, at least close to the
transition, the emergent microstructure depends on the composition and
compressibility of the melt. Hexagonal structures, as well as lamellae or
random structures with a unique wavelength, can be energetically favorable.Comment: 19 pages, 10 figures. Submitted to the Journal of Chemical Physics
(http://jcp.aip.org
Eccentric connectivity index
The eccentric connectivity index is a novel distance--based molecular
structure descriptor that was recently used for mathematical modeling of
biological activities of diverse nature. It is defined as \,, where and
denote the vertex degree and eccentricity of \,, respectively. We survey
some mathematical properties of this index and furthermore support the use of
eccentric connectivity index as topological structure descriptor. We present
the extremal trees and unicyclic graphs with maximum and minimum eccentric
connectivity index subject to the certain graph constraints. Sharp lower and
asymptotic upper bound for all graphs are given and various connections with
other important graph invariants are established. In addition, we present
explicit formulae for the values of eccentric connectivity index for several
families of composite graphs and designed a linear algorithm for calculating
the eccentric connectivity index of trees. Some open problems and related
indices for further study are also listed.Comment: 25 pages, 5 figure
Algorithms for Computing Wiener Indices of Acyclic and Unicyclic Graphs
Let be a molecular graph, where and are the
sets of vertices (atoms) and edges (bonds). A topological index of a molecular
graph is a numerical quantity which helps to predict the chemical/physical
properties of the molecules. The Wiener, Wiener polarity and the terminal
Wiener indices are the distance based topological indices. In this paper, we
described a linear time algorithm {\bf(LTA)} that computes the Wiener index for
acyclic graphs and extended this algorithm for unicyclic graphs. The same
algorithms are modified to compute the terminal Wiener index and the Wiener
polarity index. All these algorithms compute the indices in time
The hyper-Wiener index of the generalized hierarchical product of graphs
AbstractThe hyper Wiener index of the connected graph G is WW(G)=12∑{u,v}⊆V(G)(d(u,v)+d(u,v)2), where d(u,v) is the distance between the vertices u and v of G. In this paper we compute the hyper-Wiener index of the generalized hierarchical product of two graphs and give some applications of this operation
Embeddability of open-ended carbon nanotubes in hypercubes
AbstractA graph that can be isometrically embedded into a hypercube is called a partial cube. An open-ended carbon nanotube is a part of hexagonal tessellation of a cylinder. In this article we determine all open-ended carbon nanotubes which are partial cubes
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