On the rigidity of molecular graphs
type: Item
instance of: scholarly article
title: On the rigidity of molecular graphs
zbMATH Open document ID: 1199.52036
author: Bill Jackson
author: Tibor Jordán
published in: Combinatorica
publication date: 2009-12-28T00:00:00Z
review text: A graph is called rigid if a generic bar-and-joint framework corresponding to it is rigid in the usual (kinematic) sense. The authors study the rigidity of squares of graphs, which are also called molecular graphs because they are used to study the flexibility of molecules. The Molecular Conjecture, posed in 1984 by \textit{T. S. Tay} and \textit{W. Whiteley} [Topologie Struct. 9, 31--38 (1984; Zbl 0541.51021)], states that the square \(G^{2}\) of a graph \(G\) of minimum degree at least two is rigid if and only if \(5G\) contains six edge-disjoint spanning trees. The main result of the paper under review is a lower bound on the degrees of freedom of \(G^{2}\) in terms of forest covers of \(G\). This implies that the existence of the above six spanning trees is a necessary condition for the rigidity of \(G^{2}\).
reviewed by: Johann Linhart
Mathematics Subject Classification ID: 52C25
Mathematics Subject Classification ID: 05C90
Mathematics Subject Classification ID: 52B40
zbMATH DE Number: 5651821
zbMATH Keywords: rigidity of graphs
zbMATH Keywords: molecular conjecture
MaRDI profile type: Publication
full work available at URL: s00493-008-2287-z
OpenAlex ID: W2043683546
cites work: Combinatorial Algorithm for a Lower Bound on Frame Rigidity
cites work: Generic rigidity of molecular graphs via ear decomposition
cites work: Q4279195
cites work: The \(d\)-dimensional rigidity matroid of sparse graphs
cites work: The Dress conjectures on rank in the 3-dimensional rigidity matroid
cites work: ON THE RANK FUNCTION OF THE 3-DIMENSIONAL RIGIDITY MATROID
cites work: Rigid components in molecular graphs
cites work: Pin-collinear body-and-pin frameworks and the molecular conjecture
cites work: Brick partitions of graphs
cites work: The generic rank of body-bar-and-hinge frameworks
cites work: Generic rigidity in three-dimensional bond-bending networks
cites work: Edge-Disjoint Spanning Trees of Finite Graphs
cites work: Linking (n-2)-dimensional panels in n-space. II: (n-2,2)-frameworks and body and Hinge structures
cites work: Q3327256
cites work: On the Problem of Decomposing a Graph into n Connected Factors
cites work: The Union of Matroids and the Rigidity of Frameworks
cites work: Q4717849
DOI: 10.1007/S00493-008-2287-Z
description: scientific article; zbMATH DE number 5651821