What is maximum energy level of molecules?
In the painting "Dream Caused by the Flight of a Bee" by Salvador Dali, among other strange items, we may see a fish flying in midair. There is also one in the drawing "Big Fish Eat Little Fish" by Bruegel the Elder. One may conclude that there is a flying fish in every picture out there. One may even publish a paper to that effect. But I believe you see the shortcomings of this way of reasoning.

Same thing here. Molecules don't have the maximum energy level. You may think of the energy required to break a  molecule, but that's not a level; rather, it is a lower bound of the continuous spectrum. You may think of the energy levels of its constituent atoms, but those are quite irrelevant to the energy levels of the molecule. Moreover, atoms don't have the maximum energy level either.

At this point you may wonder: what were the authors thinking? Were they (together with the editors) just lunatics? Not at all! Then what did they have in mind? Why, they said it loud and clear: the spectra of graphs. How are those related to the spectra of molecules? Is a molecule the same as a graph? Of course it isn't... unless we are looking at it within certain simplistic approximation where it is.

Yes, this is the old good Hückel method once again. You take a conjugated $\pi$ system, draw a graph of the molecule, solve it for eigenvalues, and get the $\pi$ energy levels of the molecule (only upside down). Naturally, there is but a finite number of those, and there inevitably is a maximum. That's what they mean when talking about the "maximum energy level of a molecule".



Now, these $\pi$ levels typically include the frontier orbitals (HOMO and LUMO), and those next to them, and answer pretty much any chemical question about the molecule. All in all, it is quite natural and easy to forget altogether about all other orbitals and energy levels of the molecule ($\sigma$ bonds and stuff). But they are still there, and what I said before is still true.

So it goes.