A note on the order of iterated line digraphs

TL;DR

This paper introduces a novel algebraic method using minimal polynomials to determine the vertex count recurrence in iterated line digraphs, demonstrated on various graph types and related to enumerating specific word sequences.

Contribution

It presents a new technique based on minimal polynomials to find recurrence relations for iterated line digraphs, expanding analytical tools in graph theory.

Findings

Method applies to cyclic Kautz, unicyclic, and acyclic digraphs.

Enables enumeration of ternary length-2 squarefree words.

Provides explicit recurrence formulas for vertex counts.

Abstract

Given a digraph $G$, we propose a new method to find the recurrence equation for the number of vertices $n_k$ of the $k$-iterated line digraph $L^k(G)$, for $k\geq0$, where $L^0(G)=G$. We obtain this result by using the minimal polynomial of a quotient digraph $\pi(G)$ of $G$. We show some examples of this method applied to the so-called cyclic Kautz, the unicyclic, and the acyclic digraphs. In the first case, our method gives the enumeration of the ternary length-2 squarefree words of any length.