Summary: 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 \(|A|\geq \aleph _{0}\). 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., 96 (2022), 1141-1157). It is denoted by \(J_{A}^{\G}\) and it is called graph Lipscomb’s space associated with the graph \(\G\) 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 \(\G\) for which \(J_{A}^{\G}\) is connected. In the particular case when \(A\) is finite, some supplementary characterizations are presented.