Rigidity of Molecular Structures: Generic and Geometric Analysis

Whiteley, Walter

doi:10.1007/0-306-47089-6_2

Cited by 30 publications

(30 citation statements)

References 21 publications

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“…The detailed calculation can be seen in [3,Proposition 3.4] or [33,Proposition 3] for the 3-dimensional case and the technique can apply to the general dimensional case without any modification.…”

Section: Proof Of the Molecular Conjecturementioning

confidence: 99%

“…As we mentioned in Introduction, in 3-dimensional space the projective dual of nonparallel paneland-hinge frameworks are "hinge-concurrent" body-and-hinge frameworks, which are also called molecular frameworks [14,33] because they are used to study the flexibility of molecules. Since the infinitesimal rigidity is invariant under the duality [3, Section 3.6] (see also [29] for more detailed descriptions on the related topic), it follows from Theorem 6.7 that a simple graph G = (V, E) can be realized as a molecular framework (G, p) which satisfies rank R(…”

Section: Proof Let K = Def( G) By Proposition 32 We Have Rank R(gmentioning

confidence: 99%

See 1 more Smart Citation

A Proof of the Molecular Conjecture

Katoh

Tanigawa

2011

Discrete Comput Geom

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A d-dimensional body-and-hinge framework is a structure consisting of rigid bodies connected by hinges in d-dimensional space. The generic infinitesimal rigidity of a body-and-hinge framework has been characterized in terms of the underlying multigraph independently by Tay and Whiteley as follows: A multigraph G can be realized as an infinitesimally rigid body-and-hinge framework by mapping each vertex to a body and each edge to a hinge if and only if d+1 2

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Section: Proof Of the Molecular Conjecturementioning

confidence: 99%

Section: Proof Let K = Def( G) By Proposition 32 We Have Rank R(gmentioning

confidence: 99%

A Proof of the Molecular Conjecture

Katoh

Tanigawa

2011

Discrete Comput Geom

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“…This conjecture was given by Tay and Whiteley in [15], and subsequently appeared in several different forms, see [18,19,20,21,22]. It is usually formulated in terms of 'body-and-hinge' frameworks, which will be described in Subsection 2.1.…”

Section: Rigid Graphs and Frameworkmentioning

confidence: 99%

“…In this paper we consider an important special case, which has been a focus of recent research: characterize when the square of a graph is rigid, where the square G 2 of a graph G is obtained from G by adding a new edge uv for each pair u, v ∈ V (G) of distance two in G, see Figure 1. Squares of graphs are sometimes called molecular graphs, because they are used to study the flexibility of molecules, particularly biomolecules such as proteins [19,22]. Franzblau [1,2] has developed combinatorial algorithms for computing lower and upper bounds on the degrees of freedom of molecular graphs using ear-decompositions of graphs.…”

Section: Introductionmentioning

confidence: 99%

On the rigidity of molecular graphs

Jackson¹,

Jordán²

2008

Combinatorica

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The rigidity of squares of graphs in three-space has important applications to the study of flexibility in molecules. The Molecular Conjecture, posed in 1984 by T-S. Tay and W. Whiteley, states that the square G 2 of a graph G of minimum degree at least two is rigid if and only if G has six spanning trees which cover each edge of G at most five times. We give a lower bound on the degrees of freedom of G 2 in terms of forest covers of G. This provides a self-contained proof that the existence of the above six spanning trees is a necessary condition for the rigidity of G 2 . In addition, we prove that the truth of the Molecular Conjecture would imply that our lower bound is tight, and would also imply that a conjecture of Jacobs on 'independent' squares is valid.

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“…It is possible to devise internal coordinate systems which use only distances as coordinates; this approach is related to distance geometry [18] and rigidity theory [36], [85]. For attempts in two dimensions to utilize both distances and angles as internal coordinates, see [88], [89]. Internal coordinate systems which are based on collections of bond lengths, bond angles, and torsion angles [70], [82], are loosely called valence coordinates by chemists, but until recently [31] there has been little effort to understand exactly which combinations of these valence coordinates form "good" internal coordinate systems.…”

Section: Introductionmentioning

confidence: 99%

Polyspherical Coordinate Systems on Orbit Spaces with Applications to Biomolecular Shape

Dix

2006

Acta Appl Math

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Abstract. A general theory of molecular internal coordinates of valence type is presented based on the concept of a Z-system. The Z-system can be considered as a discrete mathematical generalization of the Z-matrix (a molecular geometry file format familiar to chemists) which avoids the principal disadvantage of Z-matrices. Z-matrices are usually only employed for small molecules because there is no easy way to glue two Z-matrices together to get the Zmatrix of a larger molecule. It is shown that Z-matrices are simply Z-systems together with additional extraneous structures and that the Z-systems for any two molecules can be naturally glued together to obtain a Z-system for the combined molecule. A general mathematical framework suitable for the detailed study of molecular geometry is introduced and applied to five and sixmembered molecular rings. A classification of shapes of hexagons with opposite sides and angles congruent is given with explicit parameterizations of the flexible and rigid solutions. The entire mathematical formalism generalizes to a theory of polyspherical coordinate systems on orbit spaces of the group of n-dimensional rigid motions acting on finite collections of points in n-dimensional Euclidean space. The n-dimensional Z-system is a new discrete structure related to abstract simplicial complexes, graded posets, and iterated line graphs. Complete proofs of all the n-dimensional results are given, and connections to other areas of mathematics are noted.

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Rigidity of Molecular Structures: Generic and Geometric Analysis

Cited by 30 publications

References 21 publications

A Proof of the Molecular Conjecture

A Proof of the Molecular Conjecture

On the rigidity of molecular graphs

Polyspherical Coordinate Systems on Orbit Spaces with Applications to Biomolecular Shape

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