Iterative Laplacian-trace characterization of maximal complexity

Prove that graphs of maximal complexity among graphs with at most n+1 vertices and at most m edges are obtained by successively imposing exact vertex count and edge count, nearly equal degrees, minimum triangle count, and, for each r≥4, maximal values of the alternating Laplacian trace objective, with the process terminating after at most n+1 steps in a class consisting only of cospectral graphs of maximal complexity.

Background

Conjecture 1 proposes a general structural procedure for identifying graphs with the maximum number of spanning trees under upper bounds on the number of vertices and edges. The proposed procedure first forces the bounds to be attained, then minimizes degree imbalance and triangle count, and subsequently optimizes alternating traces of powers of the Laplacian.

The results proved earlier in the paper establish maximal complexity only over the nearly regular subclass, and the authors state that extending the conclusion to the larger class of all admissible graphs would validate the conjectural structure. Computational evidence for graphs with at most nine vertices and an asymptotic heuristic are offered as support, but the conjecture remains unresolved.