1,182 research outputs found
The Vertex Version of Weighted Wiener Number for Bicyclic Molecular Structures
Graphs are used to model chemical compounds and drugs. In the graphs, each vertex represents an atom of molecule and edges between the corresponding vertices are used to represent covalent bounds between atoms. We call such a graph, which is derived from a chemical compound, a molecular graph. Evidence shows that the vertex-weighted Wiener number, which is defined over this molecular graph, is strongly correlated to both the melting point and boiling point of the compounds. In this paper, we report the extremal vertex-weighted Wiener number of bicyclic molecular graph in terms of molecular structural analysis and graph transformations. The promising prospects of the application for the chemical and pharmacy engineering are illustrated by theoretical results achieved in this paper
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
On the Randi\'{c} index and conditional parameters of a graph
The aim of this paper is to study some parameters of simple graphs related
with the degree of the vertices. So, our main tool is the matrix
whose ()-entry is where denotes the degree of the vertex . We study
the Randi\'{c} index and some interesting particular cases of conditional
excess, conditional Wiener index, and conditional diameter. In particular,
using the matrix or its eigenvalues, we obtain tight bounds on the
studied parameters.Comment: arXiv admin note: text overlap with arXiv:math/060243
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