More on the Zagreb indices inequality

Abstract

The Zagreb indices are very popular topological indices in mathematical chemistry and attracted a lot of attention in recent years. The first and second Zagreb indices of a graph G = (V, E) are defined as M 1 (G) = vi∈V d 2 ı and M 2 (G) = υi∼υȷ (didȷ), where di denotes the degree of a vertex υi and υi ∼ υȷ represents the adjacency of vertices υi and υȷ in G. It has been conjectured that M 1 /n ≤ M 2 /m holds for a connected graph G with n = |V | and m = |E|. Later, it is proved that this inequality holds for some classes of graphs but does not hold in general. This inequality is proved to be true for graphs with di ∈, where h ≥ z(z − 1)/2. In this paper, we prove that the graphs satisfy the inequality if the sequences (di) and (Si) have the similar monotonicity, where Si = υȷ∈N (υi) dȷ and N (υi) = {υȷ ∈ V |υi ∼ υȷ}. As a consequence, we present an infinite family of connected graphs with di ∈ [1, ∞), for which the inequality holds. Moreover, we establish the relations between M 1 /n and M 2 /m in case of general graphs.

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