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IMPROVED ALGORITHMS FOR ENUMERATING TREELIKE CHEMICAL GRAPHS WITH GIVEN PATH FREQUENCY
"... This paper considers the problem of enumerating all nonisomorphic treelike chemical graphs with given path frequency, where “treelike ” means that the graph can be viewed as a tree if multiple edges (i.e., edges with the same end points) and a benzene ring are treated as one edge and one vertex, ..."
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This paper considers the problem of enumerating all nonisomorphic treelike chemical graphs with given path frequency, where “treelike ” means that the graph can be viewed as a tree if multiple edges (i.e., edges with the same end points) and a benzene ring are treated as one edge and one vertex, respectively, and “path frequency ” is a vector of the numbers of specified vertexlabeled paths that must be realized in every output. This and related problems have several potential applications such as classification of chemical compounds, structure determination using massspectrum and/or NMR and design of novel chemical compounds. For this problem, several studies have been done. Recently, Fujiwara et al. (2008) showed two formulations and for each of them, they gave a branchandbound algorithm, which combined efficient enumeration of nonisomorphic trees with bounding operations based on the path frequency and the atomatom bonds to avoid the generation of invalid trees. In this paper, based on their work and a result of Nagamochi (2006), we introduce two new bounding operations, the detachmentcut and the Hcut, to further reduce the size of the search space. We performed computational experiments to compare our proposed algorithms with those of Fujiwara et al. (2008) using some chemical compound data obtained from the KEGG LIGAND database
Discrete Mathematics for Combinatorial Chemistry
, 1998
"... The aim is a description of discrete mathematics used in a project devoted to the implementation of a software package for the simulation of combinatorial chemistry. ..."
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The aim is a description of discrete mathematics used in a project devoted to the implementation of a software package for the simulation of combinatorial chemistry.
Constant Time Generation of Trees with Degree Bounds
"... Given a number n of vertices, a lower bound d on the diameter, and a capacity function ∆(k) ≥ 2, k = 0, 1,..., ⌊n/2⌋, we consider the problem of enumerating all unrooted trees T with exactly n vertices and a diameter at least d such that the degree of each vertex with distance k from the center of ..."
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Given a number n of vertices, a lower bound d on the diameter, and a capacity function ∆(k) ≥ 2, k = 0, 1,..., ⌊n/2⌋, we consider the problem of enumerating all unrooted trees T with exactly n vertices and a diameter at least d such that the degree of each vertex with distance k from the center of T is at most ∆(k). We give an algorithm that generates all such trees without duplication in O(1)time delay per output in the worst case using O(n) space. For example, our result implies that all alkanes, structural isomers of chemical graph CnH2n+2 can be generated in O(1)time delay without duplication. 1
Counting Free Binary Trees Admitting a Given Height
"... Recursive equations are derived for the exact number t h of nonisomorphic free trees which have some rooting as a binary tree of height h. Numerical results are calculated using these formulae. 1. Introduction A binary tree T can be defined as a rooted tree in which each node has degree at most 3, ..."
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Recursive equations are derived for the exact number t h of nonisomorphic free trees which have some rooting as a binary tree of height h. Numerical results are calculated using these formulae. 1. Introduction A binary tree T can be defined as a rooted tree in which each node has degree at most 3, except that the root has degree at most 2. The height of T is the maximum distance from the root node to an endnode. Binary trees are much used in theoretical computer science, with height often being a key parameter directly related to the efficiency of associated algorithms. A free binary tree F is an unrooted tree which has a node u (not necessarily unique) such that F is a binary tree when rooted at u. Our purpose is to derive formulae for the number of unlabeled free binary trees which have a rooting that produces a binary tree of height h; we say that such a tree admits height h. In general our terminology follows [3]. Unlabeled counting does not distinguish between versions of a tree...
On Rotation Distance Between Binary Coupling Trees and Applications for 3njCoefficients
, 1999
"... Generalized recoupling coefficients or 3njcoefficients for a Lie algebra (with su(2), the Lie algebra for the quantum theory of angular momentum, as generic example) can always be expressed as multiple sums over products of Racah coefficients (i.e. 6jcoefficients). In general there exist many s ..."
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Generalized recoupling coefficients or 3njcoefficients for a Lie algebra (with su(2), the Lie algebra for the quantum theory of angular momentum, as generic example) can always be expressed as multiple sums over products of Racah coefficients (i.e. 6jcoefficients). In general there exist many such expressions; we say that such an expression is optimal if the number of Racah coefficients in such a product (and, correlated, the number of summation indices) is minimal. The problem of finding an optimal expression for a given 3njcoefficient is equivalent to finding a shortest path in a graph Gn . The vertices of this graph Gn consist of binary coupling trees, representing the coupling schemes in the bra/kets of the 3njcoefficients. This is the graph of rooted (unordered) binary trees with labelled leaves, and has order (2n \Gamma 1)!!. As the order increases so rapidly, finding a shortest path is computationally achievable only for n ! 11. We present some mathematical tools ...
the Italian Leonardo Fibonacci (∼1170–∼1250), the French Blaise Pascal
, 2004
"... Abstract. This article is not a research paper, but a little note on the history of combinatorics: We present here a tentative short biography of Henri Delannoy, and a survey of his most notable works. This answers to the question raised in the title, as these works are related to lattice paths enum ..."
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Abstract. This article is not a research paper, but a little note on the history of combinatorics: We present here a tentative short biography of Henri Delannoy, and a survey of his most notable works. This answers to the question raised in the title, as these works are related to lattice paths enumeration, to the socalled Delannoy numbers, and were the first general way to solve Ballotlike problems. This version corresponds to an update (May 2002) of the abstract submitted (February 2002) by the first author to the 5th lattice path combinatorics and discrete distributions
Hiroshi Nagamochi ∗
"... In this paper, we consider an arbitrary class H of rooted graphs such that each biconnected component is given by a representation with reflectional symmetry, which allows a rooted graph to have several different representations, called embeddings. We give a general framework to design algorithms fo ..."
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In this paper, we consider an arbitrary class H of rooted graphs such that each biconnected component is given by a representation with reflectional symmetry, which allows a rooted graph to have several different representations, called embeddings. We give a general framework to design algorithms for enumerating embeddings of all graphs in H without repetition. The framework yields an efficient enumeration algorithm for a class H if the class B of biconnected graphs used in the graphs in H admits an efficient enumeration algorithm. For example, for the class B of rooted cycles, we can easily design an algorithm of enumerating rooted cycles so that delivers the difference between two consecutive cycles in constant time in a series of all outputs. Hence our framework implies that, for the class H of all rooted cacti, there is an algorithm that enumerates each cactus in constant time. 1