Note on the Comparison of the First and Second Normalized Zagreb Eccentricity Indices

The conjecture Σu∈V(G) dG(u)2 / n(G) ≤Σ uv∈E(G) dG(u)dG(v) / m(G) that compares normalized Zagreb indices attracted recently a lot of attention1–9. In this paper we analyze analogous statement in which degree dG(u) of vertex u is replaced by its eccentricityΣG(u) in which way we define novel first and second Zagreb eccentricity indices. We show that Σ u∈V(G) ΣG(u)2 / n(G) ≥Σuv∈E(G) ΣG(u)ΣG(v) / m(G) holds for all acyclic and unicyclic graphs and that neither this nor the opposite inequality holds for all bicyclic graphs.