4,637 research outputs found
GTI-space : the space of generalized topological indices
A new extension of the generalized topological indices (GTI) approach is carried out torepresent 'simple' and 'composite' topological indices (TIs) in an unified way. Thisapproach defines a GTI-space from which both simple and composite TIs represent particular subspaces. Accordingly, simple TIs such as Wiener, Balaban, Zagreb, Harary and Randićconnectivity indices are expressed by means of the same GTI representation introduced for composite TIs such as hyper-Wiener, molecular topological index (MTI), Gutman index andreverse MTI. Using GTI-space approach we easily identify mathematical relations between some composite and simple indices, such as the relationship between hyper-Wiener and Wiener index and the relation between MTI and first Zagreb index. The relation of the GTI space with the sub-structural cluster expansion of property/activity is also analysed and some routes for the applications of this approach to QSPR/QSAR are also given
Moments in graphs
Let be a connected graph with vertex set and a {\em weight function}
that assigns a nonnegative number to each of its vertices. Then, the
{\em -moment} of at vertex is defined to be
M_G^{\rho}(u)=\sum_{v\in V} \rho(v)\dist (u,v) , where \dist(\cdot,\cdot)
stands for the distance function. Adding up all these numbers, we obtain the
{\em -moment of }: M_G^{\rho}=\sum_{u\in
V}M_G^{\rho}(u)=1/2\sum_{u,v\in V}\dist(u,v)[\rho(u)+\rho(v)]. This
parameter generalizes, or it is closely related to, some well-known graph
invariants, such as the {\em Wiener index} , when for every
, and the {\em degree distance} , obtained when
, the degree of vertex . In this paper we derive some
exact formulas for computing the -moment of a graph obtained by a general
operation called graft product, which can be seen as a generalization of the
hierarchical product, in terms of the corresponding -moments of its
factors. As a consequence, we provide a method for obtaining nonisomorphic
graphs with the same -moment for every (and hence with equal mean
distance, Wiener index, degree distance, etc.). In the case when the factors
are trees and/or cycles, techniques from linear algebra allow us to give
formulas for the degree distance of their product
Automated Identification and Classification of Stereochemistry: Chirality and Double Bond Stereoisomerism
Stereoisomers have the same molecular formula and the same atom connectivity
and their existence can be related to the presence of different
three-dimensional arrangements. Stereoisomerism is of great importance in many
different fields since the molecular properties and biological effects of the
stereoisomers are often significantly different. Most drugs for example, are
often composed of a single stereoisomer of a compound, and while one of them
may have therapeutic effects on the body, another may be toxic. A challenging
task is the automatic detection of stereoisomers using line input
specifications such as SMILES or InChI since it requires information about
group theory (to distinguish stereoisomers using mathematical information about
its symmetry), topology and geometry of the molecule. There are several
software packages that include modules to handle stereochemistry, especially
the ones to name a chemical structure and/or view, edit and generate chemical
structure diagrams. However, there is a lack of software capable of
automatically analyzing a molecule represented as a graph and generate a
classification of the type of isomerism present in a given atom or bond.
Considering the importance of stereoisomerism when comparing chemical
structures, this report describes a computer program for analyzing and
processing steric information contained in a chemical structure represented as
a molecular graph and providing as output a binary classification of the isomer
type based on the recommended conventions. Due to the complexity of the
underlying issue, specification of stereochemical information is currently
limited to explicit stereochemistry and to the two most common types of
stereochemistry caused by asymmetry around carbon atoms: chiral atom and double
bond. A Webtool to automatically identify and classify stereochemistry is
available at http://nams.lasige.di.fc.ul.pt/tools.ph
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