The Linkage Problem for Group-labelled Graphs

This thesis aims to extend some of the results of the Graph Minors Project of Robertson and Seymour to "group-labelled graphs". Let $\Gamma$ be a group. A $\Gamma$-labelled graph is an oriented graph with its edges labelled from $\Gamma$, and is thus a generalization of a signed graph. Our primary result is a generalization of the main result from Graph Minors XIII. For any finite abelian group $\Gamma$, and any fixed $\Gamma$-labelled graph $H$, we present a polynomial-time algorithm that determines if an input $\Gamma$-labelled graph $G$ has an $H$-minor. The correctness of our algorithm relies on much of the machinery developed throughout the graph minors papers. We therefore hope it can serve as a reasonable introduction to the subject. Remarkably, Robertson and Seymour also prove that for any sequence $G_1, G_2, \dots$ of graphs, there exist indices $i<j$ such that $G_i$ is isomorphic to a minor of $G_j$. Geelen, Gerards and Whittle recently announced a proof of the analogous result for $\Gamma$-labelled graphs, for $\Gamma$ finite abelian. Together with the main result of this thesis, this implies that membership in any minor closed class of $\Gamma$-labelled graphs can be decided in polynomial-time. This also has some implications for well-quasi-ordering certain classes of matroids, which we discuss

Recipe theorems for polynomial invariants on ribbon graphs with half-edges

We provide recipe theorems for the Bollob\`as and Riordan polynomial $\mathcal{R}$ defined on classes of ribbon graphs with half-edges introduced in arXiv:1310.3708[math.GT]. We also define a generalized transition polynomial $Q$ on this new category of ribbon graphs and establish a relationship between $Q$ and $\mathcal{R}$.Comment: 24 pages, 14 figure

Bicircular signed-graphic matroids

Several matroids can be defined on the edge set of a graph. Although historically the cycle matroid has been the most studied, in recent times, the bicircular matroid has cropped up in several places. A theorem of Matthews from late 1970s gives a characterization of graphs whose bicircular matroids are graphic. We give a characterization of graphs whose bicircular matroids are signed-graphic.Comment: 8 page

Knot Graphs

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph which is reducible by some finite sequence of these moves, to a graph with no edges is called a knot graph. We show that the class of knot graphs strictly contains the set of delta-wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes

On perturbations of highly connected dyadic matroids

Geelen, Gerards, and Whittle [3] announced the following result: let $q = p^k$ be a prime power, and let $\mathcal{M}$ be a proper minor-closed class of $\mathrm{GF}(q)$-representable matroids, which does not contain $\mathrm{PG}(r-1,p)$ for sufficiently high $r$. There exist integers $k, t$ such that every vertically $k$-connected matroid in $\mathcal{M}$ is a rank-$(\leq t)$ perturbation of a frame matroid or the dual of a frame matroid over $\mathrm{GF}(q)$. They further announced a characterization of the perturbations through the introduction of subfield templates and frame templates. We show a family of dyadic matroids that form a counterexample to this result. We offer several weaker conjectures to replace the ones in [3], discuss consequences for some published papers, and discuss the impact of these new conjectures on the structure of frame templates.Comment: Version 3 has a new title and a few other minor corrections; 38 pages, including a 6-page Jupyter notebook that contains SageMath code and that is also available in the ancillary file

Classical Ising model test for quantum circuits

We exploit a recently constructed mapping between quantum circuits and graphs in order to prove that circuits corresponding to certain planar graphs can be efficiently simulated classically. The proof uses an expression for the Ising model partition function in terms of quadratically signed weight enumerators (QWGTs), which are polynomials that arise naturally in an expansion of quantum circuits in terms of rotations involving Pauli matrices. We combine this expression with a known efficient classical algorithm for the Ising partition function of any planar graph in the absence of an external magnetic field, and the Robertson-Seymour theorem from graph theory. We give as an example a set of quantum circuits with a small number of non-nearest neighbor gates which admit an efficient classical simulation.Comment: 17 pages, 2 figures. v2: main result strengthened by removing oracular settin