This paper proposes a new method for detection of graph isomorphism using the concept of quadratic form. Graphs/kinematic chains are represented first by quadratic form, and the comparison of two graphs is thus reduced to the comparison of two quadratic form expressions. If both the lengths and the directions of the semiaxes of quadric surfaces, which are characterized by the eigenvalues and eigenvectors, are the same, the associated graphs/kinematic chains are isomorphic. An algorithm is developed based on this idea, and tested for the counter-examples known to other methods.

1.
Dobrjanskyi
,
L.
, and
Freudenstein
,
F.
,
1967
, “
Some Applications of Graph Theory to the Structural Analysis of Mechanisms
,”
ASME J. Ind.
,
89B
, pp.
153
158
.
2.
Mruthyunjaya
,
T. S.
, and
Raghavan
,
M. R.
,
1979
, “
Structural Analysis of Kinematic Chains and Mechanisms Based on Matrix Representation
,”
ASME J. Mech. Des.
,
101
, pp.
488
494
.
3.
Zhang
,
W. J.
, and
Li
,
Q.
,
1999
, “
On a New Approach to Mechanism Topology Identification
,”
ASME J. Mech. Des.
,
121
, pp.
57
64
.
4.
Randic
,
M.
,
1974
, “
On the Recognition of Identical Graphs Representing Molecular Topology
,”
J. Chem. Phys.
,
60
, pp.
3920
3928
.
5.
Uicker
,
J. J.
, and
Raicu
,
A.
,
1975
, “
A Method for the Identification and Recognition of Equivalence of Kinematic Chains
,”
Mech. Mach. Theory
,
10
, pp.
375
383
.
6.
Yan
,
H. S.
, and
Hall
,
A. S.
,
1981
, “
Linkage Characteristic Polynomials: Definitions, Coefficients by Inspection
,”
ASME J. Mech. Des.
,
103
, pp.
578
584
.
7.
Shah
,
Y. J.
,
Davida
,
G. I.
, and
McCarthy
,
M. K.
,
1974
, “
Optimum Features and Graph isomorphism
,”
IEEE Trans. Syst. Man Cybern.
,
4
, pp.
313
319
.
8.
Ambekar
,
A. G.
, and
Agrawal
,
V. P.
,
1987
, “
Canonical Numbering of Kinematic Chains and Isomorphism Problem: min Code
,”
Mech. Mach. Theory
,
22
, pp.
453
461
.
9.
Tang
,
C. S.
, and
Liu
,
T.
,
1993
, “
The Degree Code—A New Mechanism Identifier
,”
ASME J. Mech. Des.
,
115
, pp.
627
630
.
10.
Luo, Y. F., Yang, T. L., and Cao, W. Q., 1991, “Identification on Spatial Kinematic Chains Using Incident Degree and Incident Degree Code,” Proc. of Eighth World Congress on the Theory of Machines and Mechanisms IFToMM’91, Prague, Czechoslovakia, pp. 999–1002.
11.
Mruthyunjava
,
T. S.
, and
Balasubramanian
,
H. R.
,
1987
, “
In Quest of a Reliable and Efficient Computational Test for Detection of Isomorphism in Kinematic Chains
,”
Mech. Mach. Theory
,
22
, pp.
131
139
.
12.
He, P. R., Zhang, W. J., and Li, Q., 2002, “Eigenvalue and Eigenvector Information of Graphs and Their Validity in Detection of Graph Isomorphism,” Proc. of the 2002 ASME DETC, DETC2002/MECH-34247, Montreal, Canada.
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