## Percolation critical polynomial as a graph invariant

## Abstract

Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in [0; 1] providing the threshold. Recent work has demonstrated that this polynomial may be generalized through a definition that can be applied on any periodic lattice. The polynomial depends on the lattice and on its decomposition into identical finite subgraphs, but once these are specified, the polynomial is essentially unique. On lattices for which the exact percolation threshold is unknown, the polynomials provide approximations for the critical probability with the estimates appearing to converge to the exact answer with increasing subgraph size. In this paper, I show how the critical polynomial can be viewed as a graph invariant like the Tutte polynomial. In particular, the critical polynomial is computed on a finite graph and may be found using the deletion-contraction algorithm. This allows calculation on a computer, and I present such results for the kagome lattice using subgraphs of up to 36 bonds. For one of these, I find the prediction p _{c} = 0:52440572:::, which differs from the numerical value, p _{c} = 0:52440503(5), by only 6:9 X 10 ^{-7}.

- Authors:

- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1399716

- Alternate Identifier(s):
- OSTI ID: 1101627

- Report Number(s):
- LLNL-JRNL-520672

Journal ID: ISSN 1539-3755; PLEEE8

- Grant/Contract Number:
- AC52-07NA27344

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

- Additional Journal Information:
- Journal Volume: 86; Journal Issue: 4; Journal ID: ISSN 1539-3755

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Scullard, Christian R. Percolation critical polynomial as a graph invariant. United States: N. p., 2012.
Web. doi:10.1103/PhysRevE.86.041131.
```

```
Scullard, Christian R. Percolation critical polynomial as a graph invariant. United States. doi:10.1103/PhysRevE.86.041131.
```

```
Scullard, Christian R. Thu .
"Percolation critical polynomial as a graph invariant". United States. doi:10.1103/PhysRevE.86.041131. https://www.osti.gov/servlets/purl/1399716.
```

```
@article{osti_1399716,
```

title = {Percolation critical polynomial as a graph invariant},

author = {Scullard, Christian R.},

abstractNote = {Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in [0; 1] providing the threshold. Recent work has demonstrated that this polynomial may be generalized through a definition that can be applied on any periodic lattice. The polynomial depends on the lattice and on its decomposition into identical finite subgraphs, but once these are specified, the polynomial is essentially unique. On lattices for which the exact percolation threshold is unknown, the polynomials provide approximations for the critical probability with the estimates appearing to converge to the exact answer with increasing subgraph size. In this paper, I show how the critical polynomial can be viewed as a graph invariant like the Tutte polynomial. In particular, the critical polynomial is computed on a finite graph and may be found using the deletion-contraction algorithm. This allows calculation on a computer, and I present such results for the kagome lattice using subgraphs of up to 36 bonds. For one of these, I find the prediction pc = 0:52440572:::, which differs from the numerical value, pc = 0:52440503(5), by only 6:9 X 10-7.},

doi = {10.1103/PhysRevE.86.041131},

journal = {Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics},

number = 4,

volume = 86,

place = {United States},

year = {2012},

month = {10}

}

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