The strong-connectivity of word-representable digraphs

TL;DR

This paper investigates the strong-connectivity properties of word-graphs, revealing a recurrence relation linking the count of such graphs to Stirling numbers of the second kind, thus connecting combinatorial partitions with graph connectivity.

Contribution

It introduces a recurrence relation for counting strongly connected word-graphs based on word length and alphabet size, bridging word partitions and digraph connectivity.

Findings

Number of strongly connected word-graphs expressed via recurrence relation

Connection established between word partitions and digraph connectivity

Recurrence relation involves Stirling numbers of the second kind

Abstract

A word-graph Gw is a digraph represented by a word w such that the vertex-set V(Gw) is the alphabet of w and the edge-set E(Gw) is determined by non-identical adjacent letter pairs in w. In this paper we study the strong-connectivity of word-graphs. Our main result is that the number of strongly connected word-graphs represented by l-words of over an n-alphabet can be expressed via a recurrence relation T(l,n) on the Stirling numbers of the second kind using a link between word partitions and digraph connectivity.