Enumerating simple paths from connected induced subgraphs

We present an exact formula for the ordinary generating series of the simple paths, also called self-avoiding walks, between any two vertices of a graph. Our formula involves a sum over the connected induced subgraphs and remains valid on weighted and directed graphs. An intermediary result of our approach provides a unifying vision that conciliates several concepts used in the literature for counting simple paths. We obtain a relation linking the Hamiltonian paths and cycles of a graph to its connected dominating sets

A General Purpose Algorithm for Counting Simple Cycles and Simple Paths of Any Length

International audienceWe describe a general purpose algorithm for counting simple cycles and simple paths of any length on a (weighted di)graph on N vertices and M edges, achieving a time complexity of O N + M + ω + ∆ |S |. In this expression, |S | is the number of (weakly) connected induced subgraphs of G on at most vertices, ∆ is the maximum degree of any vertex and ω is the exponent of matrix multiplication. We compare the algorithm complexity both theoretically and experimentally with most of the existing algorithms for the same task. These comparisons show that the algorithm described here is the best general purpose algorithm for the class of graphs where (ω−1 ∆ −1 +1)|S | ≤ |Cycle |, with |Cycle | the total number of simple cycles of length at most , including backtracks and self-loops. On Erdős-Rényi random graphs, we find empirically that this happens when the edge probability is larger than circa 4/N. In addition, we show that some real-world networks also belong to this class. Finally, the algorithm permits the enumeration of simple cycles and simple paths on networks where vertices are labeled from an alphabet on n letters with a time complexity of O N + M + n ω + ∆ |S |. A Matlab implementation of the algorithm proposed here is available for download

Understanding Liquid Dynamics using the Van Hove function from Inelastic Neutron Scattering Measurements

Liquid state physics remains relatively unexplored compared to solid-state physics, which achieved massive progress over the last century. The theoretical and experimental methodologies used in solid-state physics are not suitable to study the liquid state due to the latter\u27s strong time dependence and the lack of periodicity in structure. The approaches based on phonon dynamics break down when phonons are over-damped and localized in liquids. The microscopic nature of atomic dynamics and many-body interactions leading to liquid state properties such as viscosity and dielectric loss in liquids remain unclear. Inelastic neutron scattering measurements were done to study the microscopic origins of the above phenomena on two liquid state systems, water and gallium, with the atomic dynamics explored in real-space and time utilizing the Van Hove function, G(r,t). Molecular Dynamics (MD) simulations were implemented to explain the experimental observations. The Local Configurational Excitation (LCE) is the fundamental excitation that changes the topology of local connectivity in liquids. The life-time of LCE () is defined as the time it takes for an atom to lose or gain a neighbor. It was proposed through MD simulations, and later verified through neutron scattering measurements that the LCE’s are the microscopic origin of viscosity in metallic liquids at high temperatures. Generalizing this study to different types of liquids is essential to obtain a universal dynamical behavior of liquids. Towards that goal, we studied the correlated dynamics of a partly covalent liquid, gallium. We show that it is possible to achieve a universal behavior for simple metallic liquids and partially covalent liquid metals. The high dielectric loss in water is one of the anomalous properties of water. The microscopic molecular mechanism leading to this property remains unclear despite decades of research. By determining the Van Hove function of water from inelastic neutron scattering measurements, we show that the origin of the high dielectric loss is a collective reorientation of water molecules and cooperative proton tunneling involving several water molecules. The results contradict the widely held beliefs that the dielectric relaxation mechanism in water involves the rotation of a single molecule and is purely diffusive in origin