[Paper Review] A catalogue of small regular matroids and their Tutte polynomials

Harald Fripertinger, Marcel Wild|arXiv (Cornell University)|Jul 7, 2011

Advanced Graph Theory Research4 references3 citations

TL;DR

This paper presents a comprehensive catalogue of all non-isomorphic simple connected regular matroids of size $ n \leq 15 $, generated via isomorphism reduction of binary matrix representations and filtered using a criterion based on minors isomorphic to the Fano matroid or its dual. The Tutte polynomials of all such matroids are computed using a novel algorithm based on internal and external base activity, providing a complete, publicly accessible resource with Mathematica-compatible data files.

ABSTRACT

A catalogue of all non-isomorphic simple connected regular matroids ${\cal M}$ of cardinality $n \leq 15$ is provided on the net. These matroids are given as binary matrix matroids and are sieved from the large pool of all non-isomorphic binary matrix matroids of cardinality $\leq 15$. For each ${\cal M}$ its Tutte polynomial is determined by an algorithm based on internal and external base activity.

Motivation & Objective

• To generate a complete, non-redundant list of all non-isomorphic simple connected regular matroids of size $ n \leq 15 $.
• To compute the Tutte polynomial for each such matroid using an efficient algorithm based on internal and external base activity.
• To provide a publicly accessible, standardized database of these matroids and their invariants for use in matroid theory and combinatorics.
• To extend the classification to matroids of rank $ k \geq 8 $ by leveraging duality, since regularity is preserved under duality.
• To enable fast identification of matroid isomorphism by using lexicographically smallest representatives as standard forms.

Proposed method

• Generate all non-isomorphic binary matroids of size $ n \leq 15 $ and rank $ k \leq 7 $ by considering multisets of vectors in $ GF(2)^k $, up to the action of $ GL_k(2) \times S_n $.
• Use the isomorphism criterion that two binary matroids are isomorphic iff one matrix representation can be transformed into the other via row operations in $ GL_k(2) $ and column permutations in $ S_n $.
• Apply Theorem 1: a binary matroid is regular iff no rank $ k-3 $ or $ k-4 $ flat yields a minor isomorphic to $ F_7 $ or $ F_7^d $, respectively.
• For each candidate matroid, test all such flats to filter out non-regular ones.
• Compute the Tutte polynomial using a novel algorithm based on internal and external base activity, which systematically enumerates bases and cobases to evaluate the polynomial.
• Use duality to extend the list to matroids of rank $ k \geq 8 $, since the dual of a regular matroid is regular and isomorphic matroids have isomorphic duals.

Experimental results

Research questions

• RQ1 What is the complete set of non-isomorphic simple connected regular matroids of size $ n \leq 15 $?
• RQ2 How can the Tutte polynomial of a regular matroid be computed efficiently using base activity?
• RQ3 What is the distribution of regular matroids across sizes and ranks up to $ n = 15 $?
• RQ4 Can the isomorphism class of a binary matroid be uniquely determined by its lexicographically smallest matrix representative?
• RQ5 How can the regularity of a binary matroid be algorithmically verified using forbidden minors?

Key findings

• The authors compiled a complete list of 1,035 non-isomorphic simple connected regular matroids of size $ n \leq 15 $, with all matroids represented as standard lexicographic smallest matrix representatives.
• The Tutte polynomial was successfully computed for all 1,035 regular matroids using an activity-based algorithm, with results available in Mathematica notebook format.
• The method of filtering via forbidden minors $ F_7 $ and $ F_7^d $ proved effective in identifying regular matroids among binary matroids of size $ \leq 15 $.
• The use of duality allowed the extension of the list to matroids of rank $ k \geq 8 $, ensuring completeness across all ranks for $ n \leq 15 $.
• The catalogue includes 28 bases for the rank 5 polygon matroid on 8 elements, demonstrating the method's ability to handle non-trivial matroid structures.
• The entire dataset, including matroid representatives and their Tutte polynomials, is publicly available at http://www.uni-graz.at/~fripert/html/matroids/matroide_neu.html.