Six topics on inscribable polytopes

Inscribability of polytopes is a classic subject but also a lively research area nowadays. We illustrate this with a selection of well-known results and recent developments on six particular topics related to inscribable polytopes. Along the way we collect a list of (new and old) open questions.Comment: 11 page

Spheres are rare

We prove that triangulations of homology spheres in any dimension grow much slower than general triangulations. Our bound states in particular that the number of triangulations of homology spheres in 3 dimensions grows at most like the power 1/3 of the number of general triangulations.Comment: 14 pages, 1 figur

Stimulated Raman adiabatic passage-like protocols for amplitude transfer generalize to many bipartite graphs

Adiabatic passage techniques, used to drive a system from one quantum state into another, find widespread application in physics and chemistry. We focus on techniques to spatially transport a quantum amplitude over a strongly coupled system, such as STImulated Raman Adiabatic Passage (STIRAP) and Coherent Tunnelling by Adiabatic Passage (CTAP). Previous results were shown to work on certain graphs, such as linear chains, square and triangular lattices, and branched chains. We prove that similar protocols work much more generally, in a large class of (semi-)bipartite graphs. In particular, under random couplings, adiabatic transfer is possible on graphs that admit a perfect matching both when the sender is removed and when the receiver is removed. Many of the favorable stability properties of STIRAP/CTAP are inherited, and our results readily apply to transfer between multiple potential senders and receivers. We numerically test transfer between the leaves of a tree, and find surprisingly accurate transfer, especially when straddling is used. Our results may find applications in short-distance communication between multiple quantum computers, and open up a new question in graph theory about the spectral gap around the value 0.Comment: 17 pages, 3 figures. v2 is made more mathematical and precise than v

Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

Let S be a flat surface of genus g with cone type singularities. Given a bipartite graph G isoradially embedded in S, we define discrete analogs of the 2^{2g} Dirac operators on S. These discrete objects are then shown to converge to the continuous ones, in some appropriate sense. Finally, we obtain necessary and sufficient conditions on the pair (S,G) for these discrete Dirac operators to be Kasteleyn matrices of the graph G. As a consequence, if these conditions are met, the partition function of the dimer model on G can be explicitly written as an alternating sum of the determinants of these 2^{2g} discrete Dirac operators.Comment: 39 pages, minor change

Optimal Tilings of Bipartite Graphs Using Self-Assembling DNA

Motivated by the recent advancements in nanotechnology and the discovery of new laboratory techniques using the Watson-Crick complementary properties of DNA strands, formal graph theory has recently become useful in the study of self-assembling DNA complexes. Construction methods based on graph theory have resulted in significantly increased efficiency. We present the results of applying graph theoretical and linear algebra techniques for constructing crossed-prism graphs, crown graphs, book graphs, stacked book graphs, and helm graphs, along with kite, cricket, and moth graphs. In particular, we explore various design strategies for these graph families in two sets of laboratory constraints