Related papers: Uniform Mixing and Association Schemes

We study 3-random-like graphs, that is, sequences of graphs in which the densities of triangles and anti-triangles converge to 1/8. Since the random graph ${\mathcal G}_{n,1/2}$ is, in particular, 3-random-like, this can be viewed as a weak…

Graph partitioning problems emerge in a wide variety of complex systems, ranging from biology to finance, but can be rigorously analyzed and solved only for a few graph ensembles. Here, an ensemble of equitable graphs, i.e. random graphs…

Consider the random process in which the edges of a graph $G$ are added one by one in a random order. A classical result states that if $G$ is the complete graph $K_{2n}$ or the complete bipartite graph $K_{n,n}$, then typically a perfect…

We prove that a generic $k$-parameter bifurcation of a dynamical system with a monoid symmetry occurs along a generalized kernel or center subspace of a particular type. More precisely, any (complementable) subrepresentation $U$ is given a…

We consider nearest neighbor weighted random walks on the $d$-dimensional box $[n]^d$ that are governed by some function $g:[0,1] \ra [0,\iy)$, by which we mean that standing at $x$, a neighbor $y$ of $x$ is picked at random and the walk…

In this paper we isolate the combinatorial property responsible (at least in part) for the computational speedups recently observed in some quantum walk algorithms. We find that continuous-time quantum walks can exploit the covering space…

The mixing time of a discrete-time quantum walk on the hypercube is considered. The mean probability distribution of a Markov chain on a hypercube is known to mix to a uniform distribution in time O(n log n). We show that the mean…

We analyze continuous-time quantum walks on necklace graphs - cyclical graphs consisting of many copies of a smaller graph (pearl). Using a Bloch-type ansatz for the eigenfunctions, we block-diagonalize the Hamiltonian, reducing the…

Every Hadamard matrix $H$ of order $n > 1$ induces a graph with $4n$ vertices, called the Hadamard graph $\Gamma(H)$ of $H$. Since $\Gamma(H)$ is a distance-regular graph with diameter $4$, it induces a $4$-class association scheme…

Let $\mathbb{D}$ be a division ring, and let ${\mathbb{D}}^{m\times n}$ be the set of $m\times n$ matrices over $\mathbb{D}$. Two matrices $A,B\in {\mathbb{D}}^{m\times n}$ are adjacent if ${\rm rank}(A-B)=1$. By the adjacency,…

Let $G$ a bipartite graph with vertex bipartition $\{A,B\}$ and let $m=|E(G)|$. An $(A,B)$-uniformly ordered labeling of $G$ is a labeling $f\colon V\rightarrow [0,2m]$ which, among other conditions, requires that there exists $\lambda\in…

We present an infinite family of finite planar graphs $\{X_n\}$ with degree at most five and such that for some constant $c > 0$, $$ \lambda_1(X_n) \geq c(\frac{\log \diam(X_n)}{\diam(X_n)})^2\,, $$ where $\lambda_1$ denotes the smallest…

In this paper we show that for any fusion $\mathcal{B}$ of an association scheme $\mathcal{A}$, the generalized Hamming scheme $H(n,\mathcal{B})$ is a nontrivial fusion of $H(n,\mathcal{A})$. We analyze the case where $\mathcal{A}$ is the…

For a given hypergraph $H$ and a vertex $v\in V(H)$, consider a random matching $M$ chosen uniformly from the set of all matchings in $H.$ In $1995,$ Kahn conjectured that if $H$ is a $d$-regular linear $k$-uniform hypergraph, the…

Breaking of equivalence between the microcanonical ensemble and the canonical ensemble, describing a large system subject to hard and soft constraints, respectively, was recently shown to occur in large random graphs. Hard constraints must…

The Szegedy quantum walk is a discrete time quantum walk model which defines a quantum analogue of any Markov chain. The long-term behavior of the quantum walk can be encoded in a matrix called the average mixing matrix, whose columns give…

Let $G = (V,E)$ be a graph on $n$ vertices and let $m^*(G)$ denote the size of a maximum matching in $G$. We show that for any $\delta > 0$ and for any $1 \leq k \leq (1-\delta)m^*(G)$, the down-up walk on matchings of size $k$ in $G$ mixes…

We present analytical treatment of quantum walks on multidimensional hyper-cycle graphs. We derive the analytical expression of the probability distribution for strong and weak decoherence regimes. Upper bound to mixing time is obtained.

A packing of two $k$-uniform hypergraphs $H_1$ and $H_2$ is a set $\{H_1', H_2'\}$ of edge-disjoint sub-hypergraphs of the complete $k$-uniform hypergraph $K_n^{(k)}$ such that $H_1'\cong H_1$ and $H_2'\cong H_2$. Whilst the problem of…

We show that if $\gS=(V,E)$ is a regular bipartite graph for which the expansion of subsets of a single parity of $V$ is reasonably good and which satisfies a certain local condition (that the union of the neighbourhoods of adjacent…