Bounds, formulas, and usefulness of graph vulnerability measures and topological indices

Determine bounds, formulas, and practical usefulness for graph vulnerability measures and topological indices, including those used to assess communication-network stability and model chemical compounds.

Background

The paper studies Hosoya polynomials of Mycielskian graphs and uses them to derive formulas for several graph vulnerability measures and topological indices, including closeness, betweenness centrality, the Wiener index, the Hyper-Wiener index, the TSZ index, and the Harary index. The conclusion places these results within the broader research direction of determining bounds, formulas, and applications for the many vulnerability measures and topological indices used in network science and mathematical chemistry.

References

Since there are many vulnerability measures and topological indices, there are open problems of finding bounds, formulas, and usefulness of them.

— Hosoya Polynomials of Mycielskian Graphs  (2503.03055 - Vaidya et al., 4 Mar 2025) in Section 5, Discussion and Conclusion

Several questions remain open. In particular, it would be of interest to determine the minimum of DRMS among connected graphs (or trees) of a given order, to strengthen the relation between DRMS and the other domination topological indices. We leave these problems for future work.

— The Domination Reciprocal Mean Square Index  (2610.10221 - Taheri-Dehkordi, 7 Oct 2026) in Conclusion