On optimal graphs embedded into paths and rings, with analysis using l₁-spheres

In this paper we study path layouts in communication networks. Stated in graph-theoretic terms, these layouts are translated into embeddings (or linear arrangements) of the vertices of a graph with N nodes onto the points 1, 2, N of the x-axis. We look for a graph with minimum diameter D^L_c(N), for which such an embedding is possible, given a bound c ou, the cutwidth of the embedding. We develop a technique to embed the nodes of such graphs into the integral lattice points in the c-dimensional li-sphere. Using this technique, we show that the minimum diameter D^L_c(N) satisfies Rc(N) ≤ D^L_c(N) ≤ 2Rc(N), where Rc(N) is the minimum radius of a c-dimensional li-sphere that conrains N points. Extensions of the results to augmented paths and ring networks are also presented. Using geometric arguments, we derive analytical bounds for Rc(N), which result in substantial improvements on some known lower and upper bounds.