The Characterization Of Graphs With Small Bicycle Spectrum

Matroids designs are defined to be matroids in which the hyperplanes all have the same size. The dual of a matroid design is a matroid with all circuits of the same size, called a dual matroid design. The connected bicircular dual matroid designs have been characterized previously. In addition, these results have been extended to connected bicircular matroids with circuits of two sizes in the case that the associated graph is a subdivision of a 3-connected graph. In this dissertation, we will use a graph theoretic approach to discuss the characterizations of bicircular matroids with circuits of two and three sizes. We will characterize the associated graph of a bicircular matroid with circuits of two sizes. Moreover, we will provide a characterization of connected bicircular matroids with circuits of three sizes in the case that the associated graph is a subdivision of a 3-connected graph. We will also investigate the circuit spectrum of bicircular matroids whose associated graphs have minimum degree at least i for k ≥ 1, and show that there exists a set of bicycles with consecutive bicycle lengths

On excluded minors of connectivity 2 for the class of frame matroids

We investigate the set of excluded minors of connectivity 2 for the class of frame matroids. We exhibit a list $\mathcal{E}$ of 18 such matroids, and show that if $N$ is such an excluded minor, then either $N \in \mathcal{E}$ or $N$ is a 2-sum of $U_{2,4}$ and a 3-connected non-binary frame matroid

The G\mathcal{G}-invariant and catenary data of a matroid

The catenary data of a matroid $M$ of rank $r$ on $n$ elements is the vector $(\nu(M;a_0,a_1,\ldots,a_r))$, indexed by compositions $(a_0,a_1,\ldots,a_r)$, where $a_0 \geq 0$,\, $a_i > 0$ for $i \geq 1$, and $a_0+ a_1 + \cdots + a_r = n$, with the coordinate $\nu (M;a_0,a_1, \ldots,a_r)$ equal to the number of maximal chains or flags $(X_0,X_1, \ldots,X_r)$ of flats or closed sets such that $X_i$ has rank $i$,\, $|X_0| = a_0$, and $|X_i - X_{i-1}| = a_i$. We show that the catenary data of $M$ contains the same information about $M$ as its $\mathcal{G}$-invariant, which was defined by H. Derksen [\emph{J.\ Algebr.\ Combin.}\ 30 (2009) 43--86]. The Tutte polynomial is a specialization of the $\mathcal{G}$-invariant. We show that many known results for the Tutte polynomial have analogs for the $\mathcal{G}$-invariant. In particular, we show that for many matroid constructions, the $\mathcal{G}$-invariant of the construction can be calculated from the $\mathcal{G}$-invariants of the constituents and that the $\mathcal{G}$-invariant of a matroid can be calculated from its size, the isomorphism class of the lattice of cyclic flats with lattice elements labeled by the rank and size of the underlying set. We also show that the number of flats and cyclic flats of a given rank and size can be derived from the $\mathcal{G}$-invariant, that the $\mathcal{G}$-invariant of $M$ is reconstructible from the deck of $\mathcal{G}$-invariants of restrictions of $M$ to its copoints, and that, apart from free extensions and coextensions, one can detect whether a matroid is a free product from its $\mathcal{G}$-invariant.Comment: 25 page