Characterization of diameter-at-least-three graphs satisfying a total-irregularity inequality

Characterize the class of all connected graphs of diameter at least 3 such that the total irregularity _t(G) exceeds the non-self-centrality number N(G) for every graph in the class.

Background

For a connected graph G, the total irregularity _t(G) measures the aggregate pairwise differences between vertex degrees, while the non-self-centrality number N(G) measures pairwise differences between vertex eccentricities. The paper records results establishing _t(G)>N(G) for certain connected graphs of diameter at least 3, including broad subclasses of trees and unicyclic graphs, but these results do not determine the full class of graphs for which the inequality always holds.

The unresolved task is to identify precisely all connected graphs with diameter at least 3 for which the strict inequality _t(G)>N(G) is guaranteed. The authors explicitly state that the corresponding problem remains generally open.

References

Although Theorems (ii) and (iii) provide a partial solution to Problem \ref{Xu-AMC-18-LB-prblm}, and Theorem (i) gives a partial solution to Problem \ref{Xu-AMC-18-UB-prblm}, both of these problems are still generally open.

— Bounds and Optimal Results for the Total Irregularity Measure  (2501.01538 - Ali et al., 2 Jan 2025) in Problem labeled Xu-AMC-18-LB-prblm, Section 4, subsection 'Lower bounds on the total irregularity'; reaffirmed in Section 5, 'Open Problems'