S − r ways on a graph-lattice

Abstract

Graph-lattice has vertices at points with non-negative integer coordinates. Each vertex has two outgoing edges: horizontal edge and vertical edge to the neighboring vertices (right and top). We considered the problem of reachability for s — r ways. s — r way consists of alternating pieces of horizontal edges or vertical edges, each piece consisting of horizontal arcs (with the exception of, perhaps, the final) has a length that is a multiple of s, and each piece consisting of vertical edges (with the exception of, perhaps, the final) has a length that is a multiple of r. We obtained formulas for the number of s — r ways, leading from the vertex to the vertex. In the second part we investigate the problem of random walks via s — r ways. The process of random walk on the s — r paths isn’t Markov process. It is shown that it is locally reduced to the Markov process on the subgraph which determined by the starting vertex.