Extremal threshold graphs for fixed order and size

Characterize in closed form which threshold graph maximizes the adjacency spectral radius among connected graphs with a prescribed number n of vertices and m edges.

Background

It is known that an adjacency-spectral-radius maximizer among connected graphs with fixed order and size is a threshold graph. This reduces the extremal problem to identifying the appropriate threshold creation sequence.

A general closed-form identification is not known. Existing progress includes metaheuristic searches and bounds based on cubic polynomials whose coefficients are derived from the creation sequence.

References

Despite this characterization being known in principle, identifying which threshold graph is extremal for given $(n,m)$ remains theoretically difficult in general.

— The Spectral Theory of Threshold Graphs: A Survey  (2610.22838 - Manna et al., 16 Oct 2026) in Section 8; Open Problem, Section 12