Canonical Labeling Research Articles

The Steiner triple systems of order 19

Using an orderly algorithm, the Steiner triple systems of order1919are classified; there are11,084,874,82911,\!084,\!874,\!829pairwise nonisomorphic such designs. For each design, the order of its automorphism group and the number of Pasch configurations it contains are recorded;2,5912,\!591of the designs are anti-Pasch. There are three main parts of the classification: constructing an initial set of blocks, the seeds; completing the seeds to triple systems with an algorithm for exact cover; and carrying out isomorph rejection of the final triple systems. Isomorph rejection is based on the graph canonical labeling softwarenautysupplemented with a vertex invariant based on Pasch configurations. The possibility of using the (strongly regular) block graphs of these designs in the isomorphism tests is utilized. The aforementioned value is in fact a lower bound on the number of pairwise nonisomorphic strongly regular graphs with parameters(57,24,11,9)(57,24,11,9).

Algebraic Nuclear Surfaces in String Space

We show by algebraic methods that the requirement that a nucleon be a representation of the space in which we live, i.e. an irreducible representation (IR) of Lorentz space, is already sufficient to guarantee a dipolar or quadrupolar shape. Then when odd-A nuclei are modeled by the tensor product, it is found that these representations lie in a Calabi-Yau or string space. In this way a nucleus is bound by strings of electric flux lines generated by the spinning protons which dispenses with the ’strong’ force. The mesons are the quanta of the string field. Additional achievements of the model are CP - invariance, canonical state labeling and a fiberation that not only introduces a Yang-Mills field but also leads to the well-known angular momentum operators for a coupled system of P protons and N neutrons which together with their dual operators constitute the six generators of O(4). Also the tensor product dictates a system of entangled nucleons in a spin lattice.

Towards the canonical tensor operators of uq(3). II. The denominator function problem

The explicit denominator (normalization) function of the canonical tensor operators of the quantum algebra uq(3), corresponding to the maximal null space case is derived ab initio in terms of double basic hypergeometric series, which cannot be obtained as any q-extension of the SU(3) denominator polynomial Gb″1(Δ,x) in terms of multiple (double or triple) balanced hypergeometric series, introduced by Biedenharn, Louck, and their collaborators (although their q=1 versions are shown being equivalent). The corresponding orthonormal seed isoscalar factors of the coupling (Wigner–Clebsch–Gordan) coefficients of uq(3) and SU(3) with multiple irreducible representations are presented. Conjectured expression of the q-polynomials [which ratios appear in the uq(3) and (new) SU(3) denominator functions for an arbitrary value of the canonical multiplicity label t of the repeating irreducible representations] in terms of multiple partition dependent q-series (extension of the maximal and minimal null space versions) is presented and considered.

Comment on “Isomorphism, Automorphism Partitioning, and Canonical Labeling Can Be Solved in Polynomial-Time for Molecular Graphs”

Comment on “Isomorphism, Automorphism Partitioning, and Canonical Labeling Can Be Solved in Polynomial-Time for Molecular Graphs”

Isomorphism, Automorphism Partitioning, and Canonical Labeling Can Be Solved in Polynomial-Time for Molecular Graphs

The graph isomorphism problem belongs to the class of NP problems, and has been conjectured intractable, although probably not NP-complete. However, in the context of chemistry, because molecules are a restricted class of graphs, the problem of graph isomorphism can be solved efficiently (i.e., in polynomial-time). This paper presents the theoretical results that for all molecules, the problems of isomorphism, automorphism partitioning, and canonical labeling are polynomial-time problems. Simple polynomial-time algorithms are also given for planar molecular graphs and used for automorphism partitioning of paraffins, polycyclic aromatic hydrocarbons (PAHs), fullerenes, and nanotubes.

Pruning the search tree in the constructive enumeration of molecular graphs

Exhaustive and nonredundant constructive enumeration of molecular graphs (vertex colored multigraphs) with possible further constraints such as prescribed valence states of atoms (degrees of each vertex for each edge multiplicity) is presented. The employed canonical labeling of vertices is based on maximum code formed from the rows of the upper-triangle part of the adjacency matrix concatenated in a sequence from the top to the bottom. The recurrent constructive scheme uses several techniques for rejection of subtrees of a search tree, such as checking conditions for sequences of valence states to be graphical and employing of the automorphism group. The description is accompanied by illustrative examples.

Determination of Topological Equivalence Classes of Atoms and Bonds in C20−C60 Fullerenes Using a New Prolog Coding Program

A new general molecular coding Prolog program is used in theoretical studies of the properties of fullerene isomers and derivatives to identify the topological equivalence classes of atoms and bonds in C 20-C60 fullerenes. The symmetry perception algorithm is based on an exhaustive search of a minimal code. During this search all canonical labeled mapping trees are generated, allowing construction of equivalence classes of atoms and bonds. Although the present algorithm makes a complete search of canonical labelings in the molecular graph, computational time is maintained within reasonable limits even for large regular graphs such as fullerenes. In the case of fullerenes, topological equivalence classes of atoms allow an easy generation of heterofullerene isomers, while the bond classes allow for the generation of addition or cycloaddition isomers. Also, the number of atom classes and the number of atoms in each class gives important information on the number of 13 C NMR signals and intensity patterns in fullerenes.

Polynomial time algorithms for recognizing and isomorphism testing of cyclic tournaments

In [3] the problem of finding an efficient criterion for isomorphism testing of cyclic graphs was posed. In the context of the theory of computational complexity the problem reduces to that of the existence of a polynomial-time algorithm for recognizing their isomorphism. The main result of the present paper is an algorithm for finding among all tournaments the cyclic ones. For cyclic tournaments generators of the automorphism group and the set of canonical labels are constructed. The running time of the algorithm is bounded by a polynomial function of the number of input tournament vertices. Thus an affirmative answer to the above problem is obtained.

An algorithm for recognizing Kekulé structures in benzenoid and coronoid systems

In order to avoid some difficulties in the peeling algorithm, Cyvin and Gutman introduced the concept of canonical labeling of the valleys for benzenoid systems. This paper presents a graph-theoretical method for determining such a labeling. This method can be generally used to recognize Kekuléan benzenoids and coronoids.

Computer generation of acyclic graphs based on local vertex invariants and topological indices. Derived canonical labelling and coding of trees and alkanes

A new procedure (GENLOIS) is presented for generating trees or certain classes of trees such as 4-trees (graphs representing alkanes), identity trees, homeomorphical irreducible trees, rooted trees, trees labelled on a certain vertex (primary, secondary, tertiary, etc.). The present method differs from previous procedures by differentiating among the vertices of a given parent graph by means of local vertex invariants (LOVIs). New graphs are efficiently generated by adding points and/or edges only to non-equivalent vertices of the parent graph. Redundant generation of graphs is minimized and checked by means of highly discriminating, recently devised topological indices based either on LOVIs or on the information content of LOVIs. All trees onN + 1 (N + 1 < 17) points could thus be generated from the complete set of trees onN points. A unique cooperative labelling for trees results as a consequence of the generation scheme. This labelling can be translated into a code for which canonical rules were recently stated by A.T. Balaban. This coding appears to be one of the best procedures for encoding, retrieving or ordering the molecular structure of trees (or alkanes).

On graph isomorphism and graph automorphism

The problem of graph isomorphism, graph automorphism and a unique graph ID is considered. A new approach to the solution of these problems is suggested. The method is based on the spectral decompositionA =ΣiλiKiof the adjacency matrixA. This decomposition is independent of the particular labeling of graph vertices, and using this decomposition one can formulate an algorithm to derive a canonical labeling of the corresponding graphG. Since the spectral decomposition uniquely determines the adjacency matrixA and hence graphG, the obtained canonical labeling can be used in order to derive a unique graph ID. In addition, if the algorithm produces several canonical labelings, all these labelings and only these labelings are connected by the elements of the graph automorphism groupG. In this way, one obtains all elements of this group. Concerning graph isomorphism, one can use a unique graph ID obtained in the above way. However, the algorithm to decide whether graphG andG′ are isomorphic can be substantially improved if this algorithm is based on the direct comparison between spectral decompositions of the corresponding adjacency matricesA andA′.

Constructive enumeration of molecular graphs with prescribed valence states

Kvasnička, V. and Pospíchal, J., 1991. Constructive enumeration of molecular graphs with prescribed valence states. Chemometrics and Intelligent Laboratory Systems, 11: 137–147. A constructive enumeration of molecular graphs (multigraphs with vertices weighted by symbols) with prescribed valence states is suggested. The graph-theoretical properties of the canonical labelling used make it possible to formulate an exhaustive and nonredundant constructive enumeration of molecular graphs. The method is illustrated by simple examples of the constructive enumeration of molecular graphs with prescribed empirical formula and valence states of atoms.

Implementation of the U(n) tensor operator calculus in a vector Bargmann Hilbert space

Using a vector coherent state framework and an associated vector Bargmann Hilbert space (1961), the structure of two classes of U(n): U(n-1) unit projective operators is shown to be intimately related to (6-j) and (9-j) coefficients of the U(n-1) subalgebra. Explicit verification of limit properties for these operators allows the unambiguous assignment of a set of canonical operator labels.

Nuclear structure on a Grassmann manifold

Products of particlelike representations of the homogeneous Lorentz group are used to construct the degrees of spin angular momentum of a composite system of protons and neutrons. If a canonical labeling system is adopted for each state, a shell structure emerges. Furthermore the use of the Dirac ring ensures that the spin is characterized by half-angles in accord with the neutron-rotation experiment. It is possible to construct a Clebsch-Gordan decomposition to reduce a state of complex angular momentum into simpler states which can be identified with α and β particles, multipole operators, etc. Finally, ground-state energy levels are calculated for all the even-even nuclei by using a differentiable manifold that is spin-graded and gauge-invariant by construction. It is shown that this manifold is Grassmann.

A technique for determining the symmetry properties of molecular graphs

AbstractA procedure has been devised to determine the automorphism partition and the automorphism group for nondirected graphs. The general approach is to reveal the permutational symmetry by means of its systematic destruction. Symmetry decomposition pathways are created by numerically weighting specific series of vertices. Each pathway leads to a discrete ordered vector. Each pair of pathways that match reveals an element of the automorphism group. Results are presented for two molecular graphs. Further potential applications of this approach to problems of isomorphism, canonical labeling, and structural indexing are discussed.

Random Graph Isomorphism

A straightforward linear time canonical labeling algorithm is shown to apply to almost all graphs (i.e. all but $o(2^{( \begin{subarray}{l} n 2 \end{subarray} )} )$) of the $2^{( \begin{subarray}{l} n 2 \end{subarray} )} $ graphs on n vertices). Hence, for almost all graphs X, any graph Y can be easily tested for isomorphism to X by an extremely naive linear time algorithm. This result is based on the following: In almost all graphs on n vertices, the largest $n^{0.15} $ degrees are distinct. In fact, they are pairwise at least $n^{0.03} $ apart.

On the Complexity of Canonical Labeling of Strongly Regular Graphs

We prove that a canonical labeling can be assigned to the n vertices of a strongly regular graph by an algorithm of $o(\exp (2n^{1/2} \log ^2 n))$ running time (in the worst case). This complexity, though still not properly subexponential, is much better than $O(2^n )$.

Notes on canonical label languages

Derivations by a phrase-structure grammar may be represented by a word over the set of indices for the rules of the grammar. The set of all label words constitutes the label language for the grammar. The canonical label language is the restriction of the label language to the set of canonical derivations of the grammar. Whether a given language may be the label language for any grammar is partially answered by restricting the question to the canonical derivations for regular and contextfree grammars.

The generalization of the Wigner-Racah angular momentum calculus, II

It is shown that for unitary unimodular groups there exists a canonical labeling scheme that is not arbitrarily but uniquely chosen by the group itself.

The generalisation of the Wigner-Racah angular momentum calculus

It is shown that for unitary unimodular groups there exists a canonical labeling scheme that is not arbitrarily but uniquely chosen by the group itself.