Note on the comparison of the first and second normalized zagreb eccentricity indices

Area of Science:

• Graph theory
• Chemical graph theory
• Mathematical chemistry

Background:

• The normalized Zagreb indices conjecture, comparing Σuv V(G) dG(u)2 / n(G) and Σuvv E(G) dG(u)dG(v) / m(G), has garnered significant recent attention.
• Vertex degrees are fundamental in graph-based chemical property analysis.

Purpose of the Study:

• To introduce and analyze novel Zagreb eccentricity indices by substituting vertex degrees with vertex eccentricities.
• To investigate the validity of an analogous inequality involving these new indices for different classes of graphs.

Main Methods:

• Definition of the first and second Zagreb eccentricity indices based on vertex eccentricities.
• Theoretical analysis of the inequality Σuv V(G) εG(u)2 / n(G) ≥ Σuvv E(G) εG(u)εG(v) / m(G) for various graph structures.
• Examination of the inequality's behavior in acyclic, unicyclic, and bicyclic graphs.

Main Results:

• The inequality Σuv V(G) εG(u)2 / n(G) ≥ Σuvv E(G) εG(u)εG(v) / m(G) is proven to hold for all acyclic and unicyclic graphs.
• It is demonstrated that neither this inequality nor its opposite holds universally for all bicyclic graphs.

Conclusions:

• The study establishes the validity of the proposed inequality for specific graph classes, expanding on existing graph index research.
• The findings highlight the distinct behavior of Zagreb eccentricity indices compared to Zagreb indices in bicyclic graphs, suggesting further research avenues.