Mechanism researchers have developed several types of codes and indices, to indicate if a pair of kinematic chains is isomorphic. Unfortunately, most of these codes or indices are either computationally inefficient or unreliable. This work establishes, for the first time, the reliability of the existing spectral techniques—characteristic polynomial and eigenvector approaches—for isomorphism detection. The reliability of characteristic polynomial of adjacency matrix is established by determining the number of pairs of non-isomorphic chains, with up to 14 links and one, two, and three degrees of freedom. The most recent eigenvector approach is critically reviewed and correct proof is provided for the statement that is the basis for this approach. It is shown, for the first time, that the eigenvector approach was able to identify all nonisomorphic chains, with up to 14 links and one, two, and three degrees of freedom. It is shown that unlike the characteristic polynomial method the eigenvector approach in worst case might take exponential time. Finally, efficient methods are suggested to the classical eigenvector approach by using the Perron–Frobenius theorem.

1.
Ambekar
,
A.
, and
Agrawal
,
V.
, 1986, “
On Canonical Numbering of Kinematic Chains and Isomorphisim Problem: MAX Code
,” ASME Mechanisms Conference.
2.
Ambekar
,
A.
, and
Agrawal
,
V.
, 1987, “
Canonical Numbering of Kinematic Chains and Isomorphism Problem: MIN Code
,”
Mech. Mach. Theory
0094-114X,
22
, pp.
453
461
.
3.
Tang
,
C.
, and
Liu
,
T.
, 1993, “
The Degree Codea New Mechanism Identifier
,”
ASME J. Mech. Des.
1050-0472,
115
, pp.
627
630
.
4.
Kim
,
J.
, and
Kwak
,
B.
, 1992, “
An Algorithm of Topological Ordering for Unique Representation of Graphs
,”
ASME J. Mech. Des.
1050-0472,
114
, pp.
103
108
.
5.
Shin
,
J.
, and
Krishnamurty
,
S.
, 1994, “
On Identification and Canonical Numbering of Pin Jointed Kinematic Chains
,”
ASME J. Mech. Des.
1050-0472,
116
, pp.
182
188
.
6.
Rao
,
A.
, and
Raju
,
D. V.
, 1991, “
Application of the Hamming Number Technique to Detect Isomorphism Among Kinematic Chains and Inversions
,”
Mech. Mach. Theory
0094-114X,
26
, pp.
55
75
.
7.
Quist
,
F.
, and
Soni
,
A.
, 1971, “
Structural Synthesis and Analysis of Kinematic Chains Using Path Matrices
,”
Proceedings of the 3rd World Congress for Theory of Machines and Mechanisms
, pp.
D213
D222
.
8.
Rao
,
A.
, and
Rao
,
C.
, 1993, “
Loop-Based Pseudo-Hamming Values. I: Testing Isomorphism and Rating Kinematic Chains
,”
Mech. Mach. Theory
0094-114X,
28
, pp.
113
127
.
9.
Yan
,
H.
, and
Hwang
,
W.
, 1985, “
Atlas of Basic Rigid Chains
,”
Proceedings of 9th Applied Mechanisms Conference
, pp.
1.1
1.8
.
10.
Yadav
,
J.
,
Pratap
,
C.
, and
Agrawal
,
V.
, 1995, “
Detection of Isomorphism Among Kinematic Chains Using the Distance Concept
,”
Mech. Mach. Theory
0094-114X,
117
, pp.
607
611
.
11.
Uicker
,
J.
, and
Raicu
,
A.
, 1975, “
A Method for Identification and Recognition of Equivalence of Kinematic Chains
,”
Mech. Mach. Theory
0094-114X,
10
, pp.
375
383
.
12.
Yan
,
H.
, and
Hwang
,
W.
, 1983, “
A Method for the Identification of Planar Linkage Chains
,”
Mech. Mach. Theory
0094-114X,
105
, pp.
658
662
.
13.
Mruthyunjaya
,
T.
, 1984, “
A Computerized Methodology for Structural Synthesis of Kinematic Chains: Part 1: Formulation
,”
Mech. Mach. Theory
0094-114X,
19
, pp.
487
495
.
14.
Mruthyunjaya
,
T.
, 1984, “
A Computerized Methodology for Structural Synthesis of Kinematic Chains: Part 2: Application to Several Fully or Partially Known Cases
,”
Mech. Mach. Theory
0094-114X,
19
, pp.
497
505
.
15.
Mruthyunjaya
,
T.
, 1984, “
A Computerized Methodology for Structural Synthesis of Kinematic Chains: Part 3: Application to New Case of 10-Link Three Freedom Chains
,”
Mech. Mach. Theory
0094-114X,
19
, pp.
507
530
.
16.
Mruthyunjaya
,
T.
, and
Balasubramanian
,
H.
, 1987, “
In Quest of a Reliable and Efficient Computational Test for Detection of Isomorphism in Kinematic Chains
,”
Mech. Mach. Theory
0094-114X,
22
, pp.
131
139
.
17.
Tsai
,
L.
, 1987, “
An Application of Linkage Characteristic Polynomial to the Topological Synthesis of Epicyclic Gear Trains
,”
ASME J. Mech., Transm., Autom. Des.
0738-0666,
109
, pp.
329
336
.
18.
Dubey
,
R.
, and
Rao
,
A.
, 1985, “
New Characteristic Polynomial: A Reliable Index to Detect Isomorphism Between Kinematic Chains
,”
Proceedings of the National Conference on Machine and Mechanism
, pp.
36
40
.
19.
Chang
,
Z.
,
Zhang
,
C.
,
Yang
,
Y.
, and
Wang
,
Y.
, 2002, “
A New Method to Mechanism Kinematic Chain Isomorphism Identification
,”
Mech. Mach. Theory
0094-114X,
37
, pp.
411
417
.
20.
He
,
P.
,
Zhang
,
W.
,
Li
,
Q.
, and
Wu
,
F.
, 2003, “
A New Method for Detection of Graph Isomorphism Based on the Quadratic Form
,”
ASME J. Mech. Des.
1050-0472,
125
, pp.
640
642
.
21.
Cubillo
,
J.
, and
Wan
,
J.
, 2005, “
Comments on Mechanism Kinematic Chain Isomorphism Identification Using Adjacent Matrices
,”
Mech. Mach. Theory
0094-114X,
40
, pp.
131
139
.
22.
Sunkari
,
R.
, and
Schmidt
,
L.
, “
Structural Synthesis of Kinematic Chains by Adapting a Mckay-Type Algorithm
,”
Mech. Mach. Theory
0094-114X,
41
, pp.
1021
1030
.
23.
Godsil
,
C.
, and
Royle
,
G.
, 2001,
Algebraic Graph Theory, Graduate Texts in Mathematics
, Vol.
207
,
Springer
,
New York
.
24.
Biggs
,
N.
, 1993,
Algebraic Graph Theory, Cambridge Mathematical Library
, Vol.
173
,
2nd ed.
,
Cambridge University Press
,
Cambridge
.
25.
Cvetkovic
,
D.
,
Rowlinson
,
P.
, and
Simic
,
S.
, 1996,
Eigenspaces of Graphs, Encyclopedia of Mathematics and its Applications
, Vol.
66
,
Cambridge University Press
,
Cambridge
.
26.
McKay
,
B.
, 1998, “
Isomorph-Free Exhaustive Generation
,”
J. Algorithms
0196-6774,
26
, pp.
306
324
.
You do not currently have access to this content.