Counting inequivalent monotone Boolean functions

Monotone Boolean functions (MBFs) are Boolean functions $f: {0,1}^n \rightarrow {0,1}$ satisfying the monotonicity condition $x \leq y \Rightarrow f(x) \leq f(y)$ for any $x,y \in {0,1}^n$. The number of MBFs in n variables is known as the $n$th Dedekind number. It is a longstanding computational challenge to determine these numbers exactly - these values are only known for $n$ at most 8. Two monotone Boolean functions are inequivalent if one can be obtained from the other by renaming the variables. The number of inequivalent MBFs in $n$ variables was known only for up to $n = 6$. In this paper we propose a strategy to count inequivalent MBF's by breaking the calculation into parts based on the profiles of these functions. As a result we are able to compute the number of inequivalent MBFs in 7 variables. The number obtained is 490013148

Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids

A prismatoid is a polytope with all its vertices contained in two parallel facets, called its bases. Its width is the number of steps needed to go from one base to the other in the dual graph. The first author recently showed that the existence of counter-examples to the Hirsch conjecture is equivalent to that of $d$-prismatoids of width larger than $d$, and constructed such prismatoids in dimension five. Here we show that the same is impossible in dimension four. This is proved by looking at the pair of graph embeddings on a 2-sphere that arise from the normal fans of the two bases.Comment: This paper merges and supersedes the papers arXiv:1101.3050 (of the last two authors) and arXiv:1102.2645 (of the first author

Colourful Simplicial Depth

Inspired by Barany's colourful Caratheodory theorem, we introduce a colourful generalization of Liu's simplicial depth. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d^2+1 and that the maximum is d^(d+1)+1. We exhibit configurations attaining each of these depths and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.Comment: 18 pages, 5 figues. Minor polishin

Expected Crossing Numbers

The expected value for the weighted crossing number of a randomly weighted graph is studied. A variation of the Crossing Lemma for expectations is proved. We focus on the case where the edge-weights are independent random variables that are uniformly distributed on [0,1].Comment: 14 page

Universal State Transfer on Graphs

A continuous-time quantum walk on a graph $G$ is given by the unitary matrix $U(t) = \exp(-itA)$, where $A$ is the Hermitian adjacency matrix of $G$. We say $G$ has pretty good state transfer between vertices $a$ and $b$ if for any $\epsilon > 0$, there is a time $t$, where the $(a,b)$-entry of $U(t)$ satisfies $|U(t)_{a,b}| \ge 1-\epsilon$. This notion was introduced by Godsil (2011). The state transfer is perfect if the above holds for $\epsilon = 0$. In this work, we study a natural extension of this notion called universal state transfer. Here, state transfer exists between every pair of vertices of the graph. We prove the following results about graphs with this stronger property: (1) Graphs with universal state transfer have distinct eigenvalues and flat eigenbasis (where each eigenvector has entries which are equal in magnitude). (2) The switching automorphism group of a graph with universal state transfer is abelian and its order divides the size of the graph. Moreover, if the state transfer is perfect, then the switching automorphism group is cyclic. (3) There is a family of prime-length cycles with complex weights which has universal pretty good state transfer. This provides a concrete example of an infinite family of graphs with the universal property. (4) There exists a class of graphs with real symmetric adjacency matrices which has universal pretty good state transfer. In contrast, Kay (2011) proved that no graph with real-valued adjacency matrix can have universal perfect state transfer. We also provide a spectral characterization of universal perfect state transfer graphs that are switching equivalent to circulants.Comment: 27 pages, 3 figure