Maximal-complexity graphs and near-regularity

Determine whether graphs of maximal complexity among all graphs with a fixed order and size must have vertex degrees differing by at most one.

Background

The paper studies graphs that maximize the number of labeled spanning trees among graphs with prescribed order and size. Its results establish this optimization only within subclasses of nearly regular graphs, whose vertex degrees differ by at most one. The authors identify whether this restriction captures all globally maximal-complexity graphs as a fundamental unresolved issue and later formulate related structural conjectures.

References

In spite of the fact that the complexity of a graph is expressed as the determinant of (any) principal minor of the Laplacian, this does not easily reveal the actual graph features that optimize complexity; embarassingly perhaps, it is still not known whether graphs of maximal complexity are found among the set of graphs whose degrees differ by at most one.

— On the extreme complexity of certain nearly regular graphs  (2502.06886 - Constantine et al., 9 Feb 2025) in Section 1, p. 1