Nonexistence conjecture for dendrimer nanostars of at least four stages

A dendrimer nanostar is the nanostar structure obtained by growing a dendrimer through a specified number of stages; a graph is a $G_\phi$-graph if it is realized as $G_\phi(A)$ for an appropriate set $A$, using the Euler $\phi$ function as in the paper. Nonexistence conjecture for dendrimer nanostars. There is no dendrimer nanostar which has grown at least four stages, as a $G_\phi$-graph. The paper proves that the two-stage nanostar $D_2$ is a $G_\phi$-graph, while reporting that attempts to prove nonexistence for nanostars grown through at least four stages have failed; the conjecture remains open.