Logical Foundations of Kinematic Chains: Graphs, Line Graphs, and Hypergraphs

1990 ◽  
Vol 112 (1) ◽  
pp. 79-83 ◽  
Author(s):  
Frank Harary ◽  
Hong-Sen Yan
Keyword(s):  
Graph Theory ◽  
Line Graph ◽  
Kinematic Chain ◽  
Line Graphs ◽  
Kinematic Chains ◽  
Logical Foundations ◽  
Unique Graph ◽  
Theory Of Graphs ◽  
Admissible Graph

In terms of concepts from the theory of graphs and hypergraphs we formulate a precise structural characterization of a kinematic chain. To do this, we require the operations of line graph, intersection graph, and hypergraph duality. Using these we develop simple algorithms for constructing the unique graph G (KC) of a kinematic chain KC and (given an admissible graph G) for forming the unique kinematic chain whose graph is G. This one-to-one correspondence between kinematic chains and a class of graphs enables the mathematical and logical power, precision, concepts, and theorems of graph theory to be applied to gain new insights into the structure of kinematic chains.

An Algorithm for Automatic Sketching of Planar Kinematic Chains

1985 ◽  
Vol 107 (1) ◽  
pp. 106-111 ◽  
Author(s):  
D. G. Olson ◽  
T. R. Thompson ◽  
D. R. Riley ◽  
A. G. Erdman
Keyword(s):  
Graph Theory ◽  
Line Graph ◽  
Kinematic Chain ◽  
Graph Representation ◽  
Synthesis Process ◽  
Line Graphs ◽  
Type Synthesis ◽  
Kinematic Chains ◽  
Synthesis Of Mechanisms ◽  
Computer Aided

One of the problems encountered in attempting to computerize type synthesis of mechanisms is that of automatically generating a computer graphics display of candidate kinematic chains or mechanisms. This paper presents the development of a computer algorithm for automatic sketching of kinematic chains as part of the computer-aided type synthesis process. Utilizing concepts from graph theory, it can be shown that a sketch of a kinematic chain can be obtained from its graph representation by simply transforming the graph into its line graph, and then sketching the line graph. The fundamentals of graph theory as they relate to the study of mechanisms are reviewed. Some new observations are made relating to graphs and their corresponding line graphs, and a novel procedure for transforming the graph into its line graph is presented. This is the basis of a sketching algorithm which is illustrated by computer-generated examples.

scholarly journals Efficient Algorithm for Generating Maximal L-Reflexive Trees

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 809
Author(s):  
Milica Anđelić ◽  
Dejan Živković
Keyword(s):  
Efficient Algorithm ◽  
Line Graph ◽  
Common Vertex ◽  
Line Graphs ◽  
Algebraic Numbers ◽  
Pisot Numbers ◽  
Full Characterization ◽  
Vertex Set ◽  
Second Largest Eigenvalue

The line graph of a graph G is another graph of which the vertex set corresponds to the edge set of G, and two vertices of the line graph of G are adjacent if the corresponding edges in G share a common vertex. A graph is reflexive if the second-largest eigenvalue of its adjacency matrix is no greater than 2. Reflexive graphs give combinatorial ground to generate two classes of algebraic numbers, Salem and Pisot numbers. The difficult question of identifying those graphs whose line graphs are reflexive (called L-reflexive graphs) is naturally attacked by first answering this question for trees. Even then, however, an elegant full characterization of reflexive line graphs of trees has proved to be quite formidable. In this paper, we present an efficient algorithm for the exhaustive generation of maximal L-reflexive trees.

Binomial incidence matrix of a semigraph

2020 ◽  
pp. 2150017
Author(s):  
Jyoti Shetty ◽  
G. Sudhakara
Keyword(s):  
Graph Theory ◽  
Complete Graph ◽  
Adjacency Matrix ◽  
Incidence Matrix ◽  
Systematic Approach ◽  
Line Graph ◽  
The Matrix ◽  
Theory Of Graphs

A semigraph, defined as a generalization of graph by  Sampathkumar, allows an edge to have more than two vertices. The idea of multiple vertices on edges gives rise to multiplicity in every concept in the theory of graphs when generalized to semigraphs. In this paper, we define a representing matrix of a semigraph [Formula: see text] and call it binomial incidence matrix of the semigraph [Formula: see text]. This matrix, which becomes the well-known incidence matrix when the semigraph is a graph, represents the semigraph uniquely, up to isomorphism. We characterize this matrix and derive some results on the rank of the matrix. We also show that a matrix derived from the binomial incidence matrix satisfies a result in graph theory which relates incidence matrix of a graph and adjacency matrix of its line graph. We extend the concept of “twin vertices” in the theory of graphs to semigraph theory, and characterize them. Finally, we derive a systematic approach to show that the binomial incidence matrix of any semigraph on [Formula: see text] vertices can be obtained from the incidence matrix of the complete graph [Formula: see text].

scholarly journals Towards a spectral theory of graphs based on the signless Laplacian, I

2009 ◽  
Vol 85 (99) ◽  
pp. 19-33 ◽  
Author(s):  
Dragos Cvetkovic ◽  
Slobodan Simic
Keyword(s):  
Graph Theory ◽  
Spectral Theory ◽  
Spectral Graph Theory ◽  
Line Graphs ◽  
Regular Graphs ◽  
Spectral Graph ◽  
Q Theory ◽  
Signless Laplacian ◽  
M Theory ◽  
Theory Of Graphs

A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M-theory. We outline a spectral theory of graphs based on the signless Laplacians Q and compare it with other spectral theories, in particular with those based on the adjacency matrix A and the Laplacian L. The Q-theory can be composed using various connections to other theories: equivalency with A-theory and L-theory for regular graphs, or with L-theory for bipartite graphs, general analogies with A-theory and analogies with A-theory via line graphs and subdivision graphs. We present results on graph operations, inequalities for eigenvalues and reconstruction problems.

scholarly journals The competition number of a generalized line graph is at most two

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Boram Park ◽  
Yoshio Sano
Keyword(s):  
Graph Theory ◽  
Necessary Conditions ◽  
Line Graph ◽  
Sufficient Conditions ◽  
Line Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
International Audience ◽  
Necessary And Sufficient ◽  
Competition Number

Graph Theory International audience In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufficient condition for the competition number of a line graph being one. In this paper, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions and sufficient conditions for the competition number of a generalized line graph being one.

scholarly journals Nonplanarity of Iterated Line Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jing Wang
Keyword(s):  
Line Graph ◽  
Crossing Number ◽  
Line Graphs ◽  
Full Characterization ◽  
Iterated Line Graph

The 1-crossing index of a graph G is the smallest integer k such that the k th iterated line graph of G has crossing number greater than 1. In this paper, we show that the 1-crossing index of a graph is either infinite or it is at most 5. Moreover, we give a full characterization of all graphs with respect to their 1-crossing index.

A More Direct Method for Structural Synthesis of Simple-Jointed Planar Kinematic Chains

1994 ◽  
Author(s):  
Varada Raju Dharanipragada ◽  
Nagaraja Kumar Yenugadhati ◽  
A. C. Rao
Keyword(s):  
Graph Theory ◽  
Computer Program ◽  
Reliable Method ◽  
Direct Method ◽  
Kinematic Chain ◽  
Degree Of Freedom ◽  
Structural Synthesis ◽  
Indirect Methods ◽  
Kinematic Chains ◽  
A Chain

Abstract Structural synthesis of kinematic chains leans heavily on indirect methods, most of them based on Graph Theory, mainly because reliable isomorphism tests are not available. Recently however, the first and third authors have established the Secondary Hamming String of a kinematic chain as an excellent indicator of its isomorphism. In the present paper this Hamming String method was applied with slight modifications for synthesizing on a PC-386, distinct kinematic chains with given number of links and family description. The computer program, written in Pascal, generated both the six-bar and all 16 eight-bar chains as well as one sample family (2008) of ten-bar chains, verifying previously established results. Hence this paper presents a direct, quick and reliable method to synthesize planar simple-jointed chains, open or closed, with single- or multi-degree of freedom, containing any number of links. A spin-off of this paper is a simple, concise and unambiguous notation for representing a chain.

On Identification and Canonical Numbering of Pin-Jointed Kinematic Chains

1994 ◽  
Vol 116 (1) ◽  
pp. 182-188 ◽  
Author(s):  
Jae Kyun Shin ◽  
S. Krishnamurty
Keyword(s):  
Graph Theory ◽  
Kinematic Chain ◽  
Simple Graphs ◽  
Robust Algorithm ◽  
Unique Representation ◽  
Kinematic Chains ◽  
Standard Code ◽  
Salient Features

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.

On the Stochastic Stability and Observability of Controlled Serial Kinematic Chains

2010 ◽  
Author(s):  
Fabio Bonsignorio
Keyword(s):  
Stochastic Stability ◽  
Stochastic System ◽  
Tangent Space ◽  
Kinematic Chain ◽  
Gaussian Distributions ◽  
Shannon Theory ◽  
Kinematic Chains ◽  
Controlled Systems ◽  
The Stability

In this paper the stability and observability of a controlled serial kinematic chain are analyzed with reference to a characterization of observability and stability for a stochastic system grounded in the application of Shannon theory to controlled systems. This approach was proposed in 2004 by H. Touchette and S. Lloyd. In particular it is analyzed in depth the case in which errors on the joints follow (concentrated) Gaussian distributions. In this case the property of Lie Groups (and related tangent space Lie algebra), studied from G. Chirikjian et al., allow to carry out the study of stochastic serial kinematic chains in a simplified way and to properly identify stability and observability conditions from a Shannon information standpoint.

scholarly journals Embedding the complement of two lines in a finite projective plane

1976 ◽  
Vol 22 (1) ◽  
pp. 27-34 ◽  
Author(s):  
Jim Totten
Keyword(s):  
Graph Theory ◽  
Projective Plane ◽  
Linear Space ◽  
Finite Projective Plane ◽  
Line Graphs ◽  
Finite Linear Space ◽  
Finite Linear

In this paper we use a result from graph theory on the characterization of the line graphs of the complete bigraphs to show that if n is any integer ≥ 2 then any finite linear space having p = n2 − n or p = n2 − n + 1 points, of which at least n2 − n have degree n + 1, and q ≤ n2 + n − 1 lines is embeddable in an FPP of order n unless n = 4. If n = 4 there is only one possible exception for each of the two values of p, and for p = n2 − n, this exception can be embedded in the FPP of order 5.

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