Nearly regularity of maximal-complexity graphs

Determine whether every graph of maximal complexity among graphs with a prescribed order and size has vertex degrees differing by at most one.

Background

The paper studies graph complexity, defined as the number of labeled spanning trees, and seeks structural properties of graphs maximizing this quantity for fixed order and size. The authors establish their optimization results only within subclasses of graphs whose vertex degrees differ by at most one, called nearly regular graphs.

They explicitly identify it as unresolved whether the global maximizers over all graphs must belong to this nearly regular class. This question motivates the later structural conjectures concerning iterative optimization by edge count, degree balance, triangle count, and higher Laplacian trace constraints.

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, page 2