Matroids with nine elements

We describe the computation of a catalogue containing all matroids with up to nine elements, and present some fundamental data arising from this cataogue. Our computation confirms and extends the results obtained in the 1960s by Blackburn, Crapo and Higgs. The matroids and associated data are stored in an online database, and we give three short examples of the use of this database.Comment: 22 page

On balanced planar graphs, following W. Thurston

Let $f:S^2\to S^2$ be an orientation-preserving branched covering map of degree $d\geq 2$, and let $\Sigma$ be an oriented Jordan curve passing through the critical values of $f$. Then $\Gamma:=f^{-1}(\Sigma)$ is an oriented graph on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced balanced planar graphs and showed that they combinatorially characterize all such $\Gamma$, where $f$ has $2d-2$ distinct critical values. We give a detailed account of this discussion, along with some examples and an appendix about Hurwitz numbers.Comment: 17 page

Enumeration of generalized polyominoes

As a generalization of polyominoes we consider edge-to-edge connected nonoverlapping unions of regular $k$-gons. For $n\le 4$ we determine formulas for the number $a_k(n)$ of generalized polyominoes consisting of $n$ regular $k$-gons. Additionally we give a table of the numbers $a_k(n)$ for small $k$ and $n$ obtained by computer enumeration. We finish with some open problems for $k$-polyominoes.Comment: 10 pages, 6 figures, 3 table

RAG: RNA-As-Graphs web resource

BACKGROUND: The proliferation of structural and functional studies of RNA has revealed an increasing range of RNA's structural repertoire. Toward the objective of systematic cataloguing of RNA's structural repertoire, we have recently described the basis of a graphical approach for organizing RNA secondary structures, including existing and hypothetical motifs. DESCRIPTION: We now present an RNA motif database based on graph theory, termed RAG for RNA-As-Graphs, to catalogue and rank all theoretically possible, including existing, candidate and hypothetical, RNA secondary motifs. The candidate motifs are predicted using a clustering algorithm that classifies RNA graphs into RNA-like and non-RNA groups. All RNA motifs are filed according to their graph vertex number (RNA length) and ranked by topological complexity. CONCLUSIONS: RAG's quantitative cataloguing allows facile retrieval of all classes of RNA secondary motifs, assists identification of structural and functional properties of user-supplied RNA sequences, and helps stimulate the search for novel RNAs based on predicted candidate motifs