Enumeration of unlabelled graphs with specified degree parities

AbstractThis paper gives a generating function for unlabelled graphs of order n. The coefficient of each monomial in this function shows the number of unlabelled graphs with given size and the number of odd vertices. Furthermore, the numerical examples are given for 1⩽n⩽9

The combinatorics of adinkras

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 67-69).Adinkras are graphical tools created to study representations of supersymmetry algebras. Besides having inherent interest for physicists, the study of adinkras has already shown nontrivial connections with coding theory and Clifford algebras. Furthermore, adinkras offer many easy-to-state and accessible mathematical problems of algebraic, combinatorial, and computational nature. In this work, we make a self-contained treatment of the mathematical foundations of adinkras that slightly generalizes the existing literature. Then, we make new connections to other areas including homological algebra, theory of polytopes, Pfaffian orientations, graph coloring, and poset theory. Selected results include the enumeration of odd dashings for all adinkraizable chromotopologies, the notion of Stiefel-Whitney classes for codes and their vanishing conditions, and the enumeration of all Hamming cube adinkras up through dimension 5.by Yan Zhang.Ph.D

Maximum size binary matroids with no AG(3,2)-minor are graphic

We prove that the maximum size of a simple binary matroid of rank $r \geq 5$ with no AG(3,2)-minor is $\binom{r+1}{2}$ and characterise those matroids achieving this bound. When $r \geq 6$, the graphic matroid $M(K_{r+1})$ is the unique matroid meeting the bound, but there are a handful of smaller examples. In addition, we determine the size function for non-regular simple binary matroids with no AG(3,2)-minor and characterise the matroids of maximum size for each rank

Quantum Theory from Principles, Quantum Software from Diagrams

This thesis consists of two parts. The first part is about how quantum theory can be recovered from first principles, while the second part is about the application of diagrammatic reasoning, specifically the ZX-calculus, to practical problems in quantum computing. The main results of the first part include a reconstruction of quantum theory from principles related to properties of sequential measurement and a reconstruction based on properties of pure maps and the mathematics of effectus theory. It also includes a detailed study of JBW-algebras, a type of infinite-dimensional Jordan algebra motivated by von Neumann algebras. In the second part we find a new model for measurement-based quantum computing, study how measurement patterns in the one-way model can be simplified and find a new algorithm for extracting a unitary circuit from such patterns. We use these results to develop a circuit optimisation strategy that leads to a new normal form for Clifford circuits and reductions in the T-count of Clifford+T circuits.Comment: PhD Thesis. Part A is 135 pages. Part B is 95 page