Characterization of bounded-degree trees satisfying the reverse total-irregularity inequality

Characterize the class of all trees with maximum degree at most 4 such that the total irregularity _t(T) is less than the non-self-centrality number N(T) for every tree in the class.

Background

The paper gives upper-bound results showing _t(T)<N(T) for several subclasses of trees, such as trees with specified diameter conditions and maximum degree at most 3, and certain maximum-degree-four trees without vertices of degree 3. These partial results do not yield a complete characterization for all trees of maximum degree at most 4.

The unresolved task is to determine exactly which maximum-degree-four trees satisfy the strict inequality _t(T)<N(T) universally within the class. The authors explicitly state that this problem remains generally open.