On d-Fibonacci digraphs

Abstract

The d-Fibonacci digraphs F(d, k), introduced here, have the number of vertices following some generalized Fibonacci-like sequences. They can be defined both as digraphs on alphabets and as iterated line digraphs. Here we study some of their nice properties. For instance, F(d, k) has diameter d + k − 2 and is semi-pancyclic; that is, it has a cycle of every length between 1 and ℓ, with ℓ ∈ {2k − 2, 2k − 1}. Moreover, it turns out that several other numbers of F(d, k) (of closed l-walks, classes of vertices, etc.) also follow the same linear recurrences as the numbers of vertices of the d-Fibonacci digraphs.

Keywords

• n-step fibonacci number, fibonacci graph, digraph on alphabet, de bruijn digraph, line digraph, adjacency matrix, spectrum