Related papers: Three Ways to Count Walks in a Digraph
Rotor walks are cellular automata that determine deterministic traversals of particles in a directed multigraph using simple local rules, yet they can generate complex behaviors. Furthermore, these trajectories exhibit statistical…
We give a simple combinatorial proof of a formula that extends a result by Grigorchuk (rediscovered by Cohen) relating cogrowth and spectral radius of random walks. Our main result is an explicit equation determining the number of `bumps'…
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…
It is demonstrated that in gate-based quantum computing architectures quantum walk is a natural mathematical description of quantum gates. It originates from field-matter interaction driving the system, but is not attached to specific qubit…
We prove that every 3-connected 2-indivisible infinite planar graph has a 1-way infinite 2-walk. (A graph is 2-indivisible if deleting finitely many vertices leaves at most one infinite component, and a 2-walk is a spanning walk using every…
In this paper, we obtain an explicit combinatorial formula for the number of solutions $(x_1,\ldots,x_r)\in \mathbb{F}_{p^{ab}}$ to the diagonal equation $x_{1}^k+\cdots+x_{r}^k=\alpha$ over the finite field $\mathbb{F}_{p^{ab}}$, with…
We study quantum walks through chains consisting of two and three star graphs. The first star has a distinguished vertex labelled START and the last has one labelled END. There are multiple paths between these two vertices, and the object…
In this paper we introduce the functional centrality as a generalization of the subgraph centrality. We propose a general method for characterizing nodes in the graph according to the number of closed walks starting and ending at the node.…
In this note we observe that a bijection related to Littelmann's root operators (for type $A_1$) transparently explains the well known enumeration by length of walks on $\N$ (left factors of Dyck paths), as well as some other enumerative…
Predicting where people can walk in a scene is important for many tasks, including autonomous driving systems and human behavior analysis. Yet learning a computational model for this purpose is challenging due to semantic ambiguity and a…
I present an algorithm that, given a number $n \geq 1$, computes a compact representation of the set of all noncrossing acyclic digraphs with $n$ nodes. This compact representation can be used as the basis for a wide range of dynamic…
We consider a discrete random walk on a diagonal lattice in two and three dimensions and obtain explicit solutions of absorption probabilities and probabilities of return in several domains. In three dimensions we consider both the cube and…
We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result extends well known enumerative formulae concerning lattice paths, and its derivation involves a classical…
Hypergraphs offer an explicit formalism to describe multibody interactions in complex systems. To connect dynamics and function in systems with these higher-order interactions, network scientists have generalised random-walk models to…
We discuss the complexity of path enumeration and counting in weighted temporal graphs. In a weighted temporal graph, each edge has an availability time, a traversal time and some real cost. We introduce two bicriteria temporal min-cost…
Network models have been widely used to study diverse systems and analyze their dynamic behaviors. Given the structural variability of networks, an intriguing question arises: Can we infer the type of system represented by a network based…
A calligraph is a graph that for almost all edge length assignments moves with one degree of freedom in the plane, if we fix an edge and consider the vertices as revolute joints. The trajectory of a distinguished vertex of the calligraph is…
A geometric graph is a graph whose vertices are points in general position in the plane and its edges are straight line segments joining these points. In this paper we give an $O(n^2 \log n)$ algorithm to compute the number of pairs of…
The diameter of a graph is the maximum distance between pairs of vertices in the graph. A pair of vertices whose distance is equal to its diameter are called diametrically opposite vertices. The collection of shortest paths between…
In the 1970s, Tutte developed a clever algebraic approach, based on certain "invariants" , to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to…