On the connectivity of graph Lipscomb's space

A central role in topological dimension theory is played by Lipscomb’s space $J_A$ since it is a universal space for metric spaces of weight $\mathrm{\mid }A\mathrm{\mid }\ge {\mathrm{\aleph }}_0$. On the one hand, Lipscomb’s space is the attractor of a possibly infinite iterated function system, i.e. it is a generalized Hutchinson-Barnsley fractal. As, on the other hand, some classical fractal sets are universal spaces, one can conclude that there exists a strong connection between topological dimension theory and fractal set theory. A generalization of Lipscomb’s space, using graphs, has been recently introduced (see R. Miculescu, A. Mihail, Graph Lipscomb’s space is a generalized Hutchinson-Barnsley fractal, Aequat. Math., $\mathbf{\text{96}}$ (2022), 1141-1157). It is denoted by $J_A^{\text{G}}$ and it is called graph Lipscomb’s space associated with the graph $\text{G}$ on the set $A$. It turns out that it is a topological copy of a generalized Hutchinson-Barnsley fractal. This paper provides a characterization of those graphs $\text{G}$ for which $J_A^{\text{G}}$ is connected. In the particular case when $A$ is finite, some supplementary characterizations are presented.