Diámetro unilateral y número de vértices en las redes New Amsterdam

Abstract

The main goal of this research is to solve the problems (∆,D*) and (∆,N)* for the case of New Amsterdam networks, where the maximum degree is ∆=2, N is the number of vertices and D* is the unilateral diameter. The problem (∆,N)* consists of finding the minimum possible unilateral diameter D* for a given number of vertices N and maximum degree ∆. For New Amsterdam digraphs, with ∆=2, the aim is to find the steps α,β,δ,γ that make the unilateral diameter minimum for this number of vertices and maximum degree. The problem (∆,D*) consists of finding the maximum number of vertices N for a given unilateral diameter D* and maximum degree ∆. For New Amsterdam digraphs, with ∆=2, the aim is to find the steps α,β,δ,γ that make the number of vertices maximum for this unilateral diameter and maximum degree. The unilateral diameter D* is the minimum between the diameter of a digraph and the diameter of its converse. The converse of a digraph is obtained changing the directions of all its arcs. A digraph is a network formed by vertices and directed edges called arcs. The diameter of a digraph is the minimum distance between two of the farthest vertices, taking into account the direction of the arcs. By definition a New Amsterdam digraph has an even number of vertices V=V_0∪V_1 with V_0={0,2,... ,N-2} and V_1={1,3,... ,N-1}, where each vertex i∈V_0 is adjacent to the vertices (mod N) i+α,i+β ∈V_1, for different odd integers α,β, and each vertex j∈V_1 is adjacent to the vertices (mod N) j+γ,j+δ ∈V_0 for odd integers γ and δ such that α+β+γ+δ≡0 (mod N).