Hierarchic graphs based on the Fibonacci numbers

Abstract

In this paper, a new class of hierarchically definable graphs are proposed and they are propersubgraphs of Hierarchic Cubic graphs. These graphs are based on the Fibonacci series by changinginitial conditions. When the initial conditions are changed, then the structure of obtained graph will bechanged. Thus, we obtained a series of hierarchically definable graphs. The obtained graphs havelogarithmic node degrees and diameters in terms of number of nodes. Thus they are comparable withincomplete hypercube graph. Sometimes, incomplete hypercube may include at least one node whosenode degree is 1. This is an unwilling case, however, the obtained graphs do not have nodes of degree 1except initial conditions graphs. Hypercube graph and hierarchic cubic network are recursively definable graphs and the obtainedgraphs are proper subgraphs of hierarchic cubic network. Thus, it is important to verify that theconstructed graphs are also recursively definable graphs. We prove that the obtained graphs are self- similar graphs or decomposable in terms of lower sized graphs in the same category.