Related papers: Excessive symmetry can preclude cutoff

It is well-known that classical random walks on regular graphs converge to the uniform distribution. Quantum walks, in their various forms, are quantizations of their corresponding classical random walk processes. Gerhardt and Watrous…

We call an oriented odd cycle alternating if it has exactly one vertex whose in-degree and out-degree are both positive. In this paper, we investigate whether certain graphs admit an orientation that avoids alternating odd cycles as…

We give new examples and describe the complete lists of all measures on the set of countable homogeneous universal graphs and $K_s$-free homogeneous universal graphs (for $s\geq 3$) that are invariant with respect to the group of all…

We consider a natural variant of the Erd\H{o}s-R\'enyi random graph process in which $k$ vertices are special and are never put into the same connected component. The model is natural and interesting on its own, but is actually inspired by…

This is a continuation of the study, begun by Ceccherini-Silberstein and Woess, of context-free pairs of groups and the related context-free graphs in the sense of Muller and Schupp. Instead of the cones (connected components with respect…

For any distribution $\pi$ with support equal to $[n] = \{1, 2,..., n \}$, we study the set $\mathcal{A}_{\pi}$ of tridiagonal stochastic matrices $K$ satisfying $\pi(i) K[i,j] = \pi(j) K[j,i]$ for all $i, j \in [n]$. These matrices…

Random walk algorithms are crucial for sampling and approximation problems in statistical physics and theoretical computer science. The mixing property is necessary for Markov chains to approach stationary distributions and is facilitated…

We prove that the family of largest cuts in the binomial random graph exhibits the following stability property: If $1/n \ll p = 1-\Omega(1)$, then, with high probability, there is a set of $n - o(n)$ vertices that is partitioned in the…

A 1992 conjecture of Alon and Spencer says, roughly, that the ordinary random graph $G_{n,1/2}$ typically admits a covering of a constant fraction of its edges by edge-disjoint, nearly maximum cliques. We show that this is not the case. The…

Let A_1,...,A_k be a collection of families of subsets of an n-element set. We say that this collection is cross-intersecting if for any i,j in [k] with i not equal to j, A in A_i and B in A_j implies that the intersection of A and B is…

Limiting distributions are derived for the sparse connected components that are present when a random graph on $n$ vertices has approximately $\half n$ edges. In particular, we show that such a graph consists entirely of trees, unicyclic…

Determining whether there exists a graph such that its crossing number and pair crossing number are distinct is an important open problem in geometric graph theory. We show that $\textit{cr}(G)=O(\mathop{\mathrm{pcr}}(G)^{3/2})$ for every…

A perfect matching of a complete graph $K_{2n}$ is a 1-regular subgraph that contains all the vertices. Two perfect matchings intersect if they share an edge. It is known that if $\mathcal{F}$ is family of intersecting perfect matchings of…

This is a chapter in a forthcoming book on completely regular codes in distance regular graphs. The chapter provides an overview, and some original results, on codes in distance regular graphs which admit symmetries via a permutation group…

Random graphs with latent geometric structure are popular models of social and biological networks, with applications ranging from network user profiling to circuit design. These graphs are also of purely theoretical interest within…

Continuous-time quantum walks on graphs is a generalization of continuous-time Markov chains on discrete structures. Moore and Russell proved that the continuous-time quantum walk on the $n$-cube is instantaneous exactly uniform mixing but…

It is known that every proper minor-closed class of graphs has bounded stack-number (a.k.a. book thickness and page number). While this includes notable graph families such as planar graphs and graphs of bounded genus, many other graph…

We show that for any natural number $s$, there is a constant $\gamma$ and a subgraph-closed class having, for any natural $n$, at most $\gamma^n$ graphs on $n$ vertices up to isomorphism, but no adjacency labeling scheme with labels of size…

Let D be an arbitrary subset of the natural numbers. For every n, let M(n;D) be the maximum of the cardinality of a set of Hamiltonian paths in the complete graph K_n such that the union of any two paths from the family contains a not…

In this thesis we introduce a new type of card shuffle called the one-sided transposition shuffle. At each step a card is chosen uniformly from the pack and then transposed with another card chosen uniformly from below it. This defines a…