2014/04/07 by Johan Kok, P.H. Fisher, Paul Fisher +9 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Graph Theory Research #Graph theory and applications #graph theory and CDMA systems #math.CO
paper · pdf · doi:10.48550/arxiv.1404.1714
13 pages. arXiv admin note: text overlap with arXiv:1404.0484
arxiv created 2014/04/07 · arxiv updated 2014/04/08
We introduce the concept of a family of finite directed graphs (order a) which are directed graphs derived from an infinite directed graph (order a), called the a-root digraph. The a-root digraph has four fundamental properties which are; V(J_∞(a)) = \vi|i ∈ \Bbb N\ and, if vj is the head of an edge (arc) then the tail is always a vertex vi, i<j and, ifvk for smallest k ∈ \Bbb N is a tail vertex then all vertices v_ℓ, k< ℓ < j are tails of arcs to vj and finally, the degree of vertex k is d(vk) = ak. The family of finite directed graphs are those limited to n ∈ \Bbb N vertices by lobbing off all vertices (and edges arcing to vertices)vt, t> n. Hence, trivially we have d(vi) ≤ ai for i ∈ \Bbb N. We present an interesting Lucassian-Zeckendorf result and other general results of interest. It is meant to be an introductory paper to encourage exploratory research.