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A differential-operator approach to the permanental polynomial
1Risk Assessment Division (7403M), Office of Pollution Prevention and Toxics, United States Environmental Protection Agency, 1200 Pennsylvania Avenue, NW, Washington, DC 20460, USA. cash.gordon@epa.gov
Journal of Chemical Information and Computer Sciences
|October 16, 2002
Summary
A new differential-operator algorithm significantly improves computation of the permanental polynomial, offering a substantial speed-up over existing methods for complex molecular structures.
Area of Science:
- Computational Chemistry
- Graph Theory
- Chemical Graph Theory
Background:
- The permanental polynomial is crucial in chemical graph theory for analyzing molecular structures.
- Existing computational methods for the permanental polynomial exhibit poor scalability, approximately O(2^n).
Purpose of the Study:
- To develop a more efficient algorithm for computing the permanental polynomial.
- To address the scalability limitations of current computational approaches.
Main Methods:
- A novel algorithm based on symbolic computation of second partial derivatives was developed.
- This differential-operator approach was applied to the permanental polynomial.
Main Results:
- The new algorithm demonstrates significantly better scalability compared to previous methods.
- For fullerene structures (n=32), performance was comparable, but for n=40, the new algorithm was over 45 times faster.
- The algorithm shows even greater relative performance improvements for polycyclic aromatic hydrocarbon structures.
Conclusions:
- The differential-operator algorithm offers a computationally superior method for determining the permanental polynomial.
- This advancement has implications for the analysis of complex molecular graphs in chemistry and related fields.
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