Canonical Numbering Research Articles

Canonical numbering systems for finite‐element codes

AbstractCanonical numbering systems are used to relate finite‐element vertices and elements to edges and faces in those elements. A numbering system is proposed that treats all the major topological element types used in practice. Also described is a set of functions that provide common evaluations of the canonical numbering data. Examples from various parts of the finite‐element analysis process are used to show the usefulness of these functions. The differences between various numbering systems used in commercial and research codes and our numbering system are described. The implementation of these functions is available as open‐source software and can be called directly from the C, C++, and Fortran languages. Published in 2009 by John Wiley & Sons, Ltd.

Algorithm for Advanced Canonical Coding of Planar Chemical Structures That Considers Stereochemical and Symmetric Information

We describe a rigorous and fast algorithm for advanced canonical coding of planar chemical structures based on the algorithm of Faulon et al. (J. Chem. Inf. Comput. Sci. 2004, 44, 427-436). Our algorithm works well even for highly symmetric structures; moreover, an advantage of our algorithm includes providing a rigorous canonical numbering of atoms with a consideration of stereochemistry and recognizing symmetric moieties. The planar structural line notation with the canonical numbering is also fit for use with stereochemical line notation. These capabilities are usable for general purposes in chemical structural coding and are particularly essential for detecting equivalent atoms in NMR studies. This algorithm was implemented on a 13C NMR chemical shift prediction system CAST/CNMR. Applications of the algorithm to several organic compounds demonstrate the practical efficiency of the rigorous coding.

Distances in Molecular Graphs

This paper discusses the finding of vertex to vertex distances in molecular graphs. Having found these distances. one can obtain a method for canonical numbering of the atoms in a molecule, which depends on the atomic properties and the distances between equivalence classes. This does not use the traditional Morgan algorithm. Using distances one can also perceive rings. Finally, substructures of interest can be detected using distances between the central atoms of various functional groups. The set of vertex distances are thus a kind of lens for examination of the graph properties of molecules. Applications have thus far been only in organic chemistry. Application to physical chemistry may appear wherever molecular graphs can be helpful, such as in calculations concerning molecules of high symmetry.

The atom assignment problem in automated de novo drug design. 2. A method for molecular graph and fragment perception.

If atom assignment onto 3D molecular graphs is to be optimized, an efficient scheme for placement must be developed. The strategy adopted in this paper is to analyze the molecular graphs in terms of cyclical and non-cyclical nodes; the latter are further divided into terminal and non-terminal nodes. Molecular fragments, from a fragments database, are described in a similar way. A canonical numbering scheme for the fragments and the local subgraph of the molecular graph enables fragments to be placed efficiently onto the molecular graph. Further optimization is achieved by placing similar fragments into bins using a hashing scheme based on the canonical numbering. The graph perception algorithm is illustrated in detail.

Chain numbering and encoding of cyclic hydrocarbons

The numbering of carbon atoms in organic molecules is called chain numbering. It is suggested that bonds be classified as chain bonds (between atoms with consecutive indices) and nonchain bonds. Among atoms we distinguish the first atom (No. 1) and the last atom (having the largest index) as well as initial and final atoms (starting and finishing unbranched sections of atomic chains). Also, internal numbering of elements within these classes is proposed. For condensed cyclic hydrocarbons, formulas for deriving the number of cycles are given, and a linear-chain encoding system is worked out; in this system, nonchain bonds are assigned the indices of their atoms. Rules for unique numbering (canonical chain numbering) are developed; they involve maximization of the indices of the initial atoms and nonchain bonds. The linear-chain encoding of cyclic hydrocarbon structures ensures unification and systematization of their names.

A new scheme for assignment of a canonical connection table

A new algorithm for deriving canonical numbering of atoms in a molecular graph has been developed. Some graph invariants, such as node properties, degree (connectivity), topological path, the smallest node ring index, etc., are encoded together to partit

Stereochemistry and sequence rules a proposal for modification of Cahn-Ingold-Prelog system

A modification of the CIP system is proposed to combine the advantages of the CIP system and the methods for computerised treatment of stereogenicity. The proposal is based strictly on the topology. The basis for ranking of ligands in the computer application is a method for the canonical numbering of the molecular graphs, therefore no transformations to the hierarchical digraphs are necessary. For the manual application of the proposed system, a transformation of rings and multiple bonds to chains which is equivalent to the canonical numbering, can be used. New terms constitutional stereogenic centre and configurational stereogenic centre are introduced to specify whether a stereogenic centre has the ligands with identical constitution or not. The term configurational stereogenic centre is a topological alternative to the geometrical term pseudoasymmetric stereogenic centre. Set of interdependent stereogenic centres is proposed as a basis for the treatment of the stereochemical differences between the ligands. The topological definition of the interdependence of the stereogenic centres is also presented. The modified system of the sequence rules consists of four rules.

On Identification and Canonical Numbering of Pin-Jointed Kinematic Chains

This paper deals with the development of a standard code for the unique representation of pin-jointed kinematic chains based on graph theory. Salient features of this method include the development of an efficient and robust algorithm for the identification of isomorphism in kinematic chains; the formulation of a unified procedure for the analysis of symmetry in kinematic chains; and the utilization of symmetry in the coding process resulting in the unique well-arranged numbering of the links. This method is not restricted to simple jointed kinematic chains only, and it can be applied to any kinematic chain which can be represented as simple graphs including open jointed and multiple jointed chains. In addition, the method is decodable as the original chain can be reconstructed unambiguously from the code values associated with the chains.

Development of a Standard Code for Colored Graphs and Its Application to Kinematic Chains

The development of an efficient solution procedure for the detection of isomorphism and canonical numbering of vertices of colored graphs is introduced. This computer-based algorithm for colored graphs is formed by extending the standard code approach developed earlier for the canonical numbering of simple noncolored graphs, which fully utilizes the capabilities of symmetry analysis of such noncolored graphs. Its application to various kinematic chains and mechanisms is investigated with the aid of examples. The method never failed to produce unique codes, and is also found to be robust and efficient. Using this method, every kinematic chain and mechanism, as well as path generators and function generators, will have their own unique codes and a corresponding canonical numbering of their respective links. Thus, based on its efficiency and applicability, this method can be used as a universal standard code for identifying isomorphisms, as well as for enumerating nonisomorphic kinematic chains and mechanisms.

An improved method of constructive enumeration of graphs

A canonical numbering of molecular graphs based on the maximum code assigned to the lower-triangle part of the adjacency matrix is used for a recurrent constructive enumeration of molecular graphs. The underlying theoretical concept - the so-called semicanonical numbering - is much more restrictive than the originally used concept of cooperative numbering. The method does not involve very time-consuming isomorphism checks of currently constructed graphs against those already constructed.

Constructing compact rectiliner planar layouts using canonical representation of planar graphs

We present a new linear-time algorithm to construct a rectilinear planar layout (horvert-representation, visibility representation) for a given planar graph. Our approach is based on the canonical representation of planar graphs and it is basically different from previous algorithms. If we direct the edges from lower-numbered vertices to higher-numbered vertices and there are n vertices out of which k vertices have out-degree greater than in-degree, then the maximum width of the constructed layout is. ∑ i=1 n max{d out(v i),0}⩽2n−4−(k−2)⩽2n−4 , and the maximum height is h + 1 ⩽ n, where h is length of the longest directed path. We discuss the selection of a good canonical numbering to be used when constructing layouts. We also show how our algorithm can be applied to compute planar layouts for planar graphs using other drawings for vertices than horizontal segments. In these layouts the drawings for vertices may have arbitrary nonequal sizes and shapes.

Topological organic chemistry. 3. Graph theory, binary and decimal adjacency matrices, and topological indices of alkanes

Adjacency matrices are used as a source of topological indices of alkanes by transformation from the binary to the decimal number system. The data are based upon a canonical numbering system of the graphs

Computer perception of constitutional (topological) symmetry: TOPSYM, a fast algorithm for partitioning atoms and pairwise relations among atoms into equivalence classes

An algorithm for the perception of constitutional symmetry in molecules (graphs) is presented, which partitions not only atoms (vertices) but also all pairwise relations among skeleton atoms into equivalence classes. The method works without canonical numbering, essentially by raising the connectivity matrix of the arbitrarily numbered molecule (graph) to its second, third, etc. power and evaluating the entries in these higher order matrices

Canonical numbering and coding of imaginary transition structures. A novel approach to the linear coding of individual organic reactions

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTCanonical numbering and coding of imaginary transition structures. A novel approach to the linear coding of individual organic reactionsShinsaku FujitaCite this: J. Chem. Inf. Comput. Sci. 1988, 28, 3, 128–137Publication Date (Print):August 1, 1988Publication History Published online1 May 2002Published inissue 1 August 1988https://doi.org/10.1021/ci00059a002RIGHTS & PERMISSIONSArticle Views24Altmetric-Citations8LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (1 MB) Get e-Alerts

Canonical numbering and coding of reaction center graphs and reduced reaction center graphs abstracted from imaginary transition structures. A novel approach to the linear coding of reaction types

Download Hi-Res ImageDownload to MS-PowerPointCite This:J. Chem. Inf. Comput. Sci. 1988, 28, 3, 137-142

Graph codes and a definition of structural similarity

A succinct linear notation system to encode the structure of a graph is exemplified. The notation requires a prior canonical numbering of the graph nodes based on the lengths of a longest path and path branches, and uses locants to designate branch positions and cyclicity. An algorithm and computer program to obtain the longest paths and a spanning tree containing a longest path is described. An index which measures the similarity of a pair of graphs is defined based on a comparison of their linear codes.

Canonical numbering of kinematic chains and isomorphism problem: min code

This paper explains the concept of minimum code and discusses its properties relevant to kinematic chains. An algorithm, based on one method available in the graph theoretic literature in chemistry, is eleborated to demonstrate its applicability in establishing minimum code of kinematic chains with simple joints. Minimum code, being unique for kinematic chains, is suitable for testing isomorphism. The decodability of this code positively indicates its possible in cataloguing (storage and retrieval) of kinematic chains and mechanisms.

Unique description of chemical structures based on hierarchically ordered extended connectivities (HOC procedures). I. Algorithms for finding graph orbits and canonical numbering of atoms

AbstractAn iterative algorithm is described for finding topological equivalence, ordering, and canonical numbering of vertexes (atoms) in molecular graphs. Like the Morgan algorithm, it is based on extended connectivities but: (i) the latter are used hierarchically, i. e., the discrimination in the next iteration is carried out only for the vertices having the same extended connectivities (ranks) at the previous iteration; (ii) at equal extended connectivities, additional discrimination is introduced by the ranks of adjacent vertices; (iii) there is no “best name” search; (iv) three levels of complexity of chemical structures are distinguished and handled by different procedures.Two schemes of application of HOC procedures are presented: one directed towards a fast canonical numbering for coding systems, and another one yielding levels of topological equivalence allowing a unique topological representation of the molecule with possible applications to similarity search, structure‐activity correlations, etc.

Unique description of chemical structures on hierarchically ordered extended connectivities (HOC procedures). II. Mathematical proofs for the HOC algorithm

AbstractThe basic notions and definitions, necessary for the better understanding of Part I of this series, are presented. The mathematical proof is given for sufficiency of the various HOC procedures for vertex canonical numbering and graph orbit finding.

Symmetry properties of chemical graphs VIII. On complementarity of isomerization modes

Isomerization mode defines the process of interconversion of one isomer into another. Several mechanisms are conceivable for degenerate rearrangements and, in general, lead to a distinctive network of relations between participating isomers. Here we consider selected modes which are complementary in the sense that if mode 1 transforms an isomer A into B, C, D etc., then mode 2 transforms the same isomer A into X, Y, Z, etc., which includes all isomers not comprised by the first mode. Physico-chemical complementarity can be translated into mathematical complementarity of associated chemical graphs. This allows us to use the tool of Graph Theory. One example of graph theoretical use is the theorem that graph G and its complement G have the same automorphism group (i.e., the same symmetry). We have shown that a close examination of a graph and its complement and their components allows us to recognize the automorphism group in some complex cases without resorting to canonical numbering or other involved procedures, and even allows us to determine isomorphism of different processes.