Recently, the first and second Zagreb indices are generalized into the variable Zagreb indices which are defined by M1(G) = ∑ u∈V (d(u))2λ and M2(G) = ∑ uv∈E (d(u)d(v)), where λ is any real number. In this paper, we prove that M1(G)/n M2(G)/m for all unicyclic graphs and all λ ∈ (−∞, 0]. And we also show that the relationship of numerical value between M1(G)/n and M2(G)/m is indefinite in the d...

We derive sharp lower bounds for the first and the second Zagreb indices (M1 and M2 respectively) for trees and chemical trees with the given number of pendent vertices and find optimal trees. M1 is minimized by a tree with all internal vertices having degree 4, while M2 is minimized by a tree where each “stem” vertex is incident to 3 or 4 pendent vertices and one internal vertex, while the res...

The first reformulated Zagreb index $EM_1(G)$ of a simple graph $G$ is defined as the sum of the terms $(d_u+d_v-2)^2$ over all edges $uv$ of $G .$ In this paper, the various upper and lower bounds for the first reformulated Zagreb index of a connected graph interms of other topological indices are obtained.