A note on Zagreb indices inequality for trees and unicyclic graphs

Abstract

For a simple graph G with n vertices and m edges, the inequality M₁(G)/n ≤ M₂(G)/m, where M₁(G) and M₂(G) are the first and the second Zagreb indices of G, is known as Zagreb indices inequality. Recently Vukičević and Graovac [VG], and Caporossi, Hansen and Vukčević [CHV] proved that this inequality holds for trees and unicyclic graphs, respectively. Here, alternative and shorter proofs of these results are presented.

[VG] D. Vukičević and A. Graovac, Comparing Zagreb M₁ and M₂ indices for acyclic molecules, MATCH Commun. Math. Comput. Chem. 57 (2007), 587-590.
[CHV] G. Caporossi, P. Hansen and D. Vukičević, Comparing Zagreb indices of cyclic graphs, MATCH Commun. Math. Comput. Chem. 63 (2010), 441-451.