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Analyzing the mixing time of random walks is a well-studied problem with applications in random sampling and more recently in graph partitioning. In this work, we present new analysis of random walks and evolving sets using more…
A periodic behavior is a well observed phenomena in biological and economical systems. We show that evolutionary games on graphs with imitation dynamics can display periodic behavior for an arbitrary choice of game theoretical parameters…
We quantize the regularity properties of classical graphs that determine spin models for singly-generated Yang-Baxter planar algebras, including the Kauffman polynomial, and construct explicit examples. A source of examples comes from…
The paper proves two theorems concerning the set of periods of periodic orbits for maps of graphs that are homotopic to the constant map and such that the vertices form a periodic orbit. The first result is that if $v$ is not a divisor of…
We investigate spectral properties of quantum graphs in the form of a periodic chain of rings with a connecting link between each adjacent pair, assuming that wave functions at the vertices are matched through conditions manifestly…
We show that the hitting time of the discrete time quantum random walk on the n-bit hypercube from one corner to its opposite is polynomial in n. This gives the first exponential quantum-classical gap in the hitting time of discrete quantum…
In the present paper, we study the continuous-time quantum walk on quotient graphs. On such graphs, there is a straightforward reduction of problem to a subspace that can be considerably smaller than the original one. Along the lines of…
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 define a class of quantum systems called regular quantum graphs. Although their dynamics is chaotic in the classical limit with positive topological entropy, the spectrum of regular quantum graphs is explicitly computable analytically…
An eigenvalue of the adjacency matrix of a graph is said to be \emph{main} if the all-1 vector is not orthogonal to the associated eigenspace. In this work, we approach the main eigenvalues of some graphs. The graphs with exactly two main…
Quantum walks have been employed widely to develop new tools for quantum information processing recently. A natural quantum walk dynamics of interacting particles can be used to implement efficiently the universal quantum computation. In…
The deep interconnection between linear algebra and graph theory allows one to interpret classical matrix invariants through combinatorial structures. To each square matrix A over a commutative ring K, one can associate a weighted directed…
Quantum walks have been useful for designing quantum algorithms that outperform their classical versions for a variety of search problems. Most of the papers, however, consider a search space containing a single marked element only. We show…
We review quantum chaos on graphs. We construct a unitary operator which represents the quantum evolution on the graph and study its spectral and wavefunction statistics. This operator is the analogue of the classical evolution operator on…
Given a symmetric $n\times n$ matrix $P$ with $0 \le P(u, v)\le 1$, we define a random graph $G_{n, P}$ on $[n]$ by independently including any edge $\{u, v\}$ with probability $P(u, v)$. For $k\ge 1$ let $\mathcal{A}_k$ be the property of…
We study continuous-time quantum walks on graphs which generalize the hypercube. The only known family of graphs whose quantum walk instantaneously mixes to uniform is the Hamming graphs with small arities. We show that quantum uniform…
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…
We consider matrices on infinite trees which are universal covers of Jacobi matrices on finite graphs. We are interested in the question of the existence of sequences of finite covers whose normalized eigenvalue counting measures converge…
In recent work, we study certain Cayley graphs associated with a finite commutative ring and their multiplicative subgroups. Among various results that we prove, we provide the necessary and sufficient conditions for such a Cayley graph to…
Quantum random walks use interference to obtain faster state space exploration, which can be used for algorithmic purposes. Photonic technologies provide a natural platform for many recent experimental demonstrations. Here we analyze…