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
\(M_1(G) =\sum_{v_i\in V} d_i^2\) and \(M_2(G) = \sum_{v_i\sim vj}
(d_i d_j)\), where \(d_i\) denotes the degree of a vertex \(v_i\) and
\(v_i \sim v_j\) represents the adjacency of vertices \(v_i\) and
\(v_j\) in \(G\). It has been conjectured that \(M_1/n \leq M_2/m\)
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 \(d_i \in [h, h + \lceil h\rceil]\) or \(d_i \in [h, h + z]\),
where \(h \geq z(z − 1)/2\). In this paper, we prove that the graphs satisfy the
inequality if the sequences (\(d_i\)) and (\(S_i\)) have the similar monotonicity,
where \(S_i = \sum_{v_j \in N(v_i)} d_j\) and \(N(v_i) = \{v_j \in V |v_i \sim v_j\}\).
As a consequence, we present an infinite family of connected graphs with
\(d_i \in [1, \infty)\), for which the inequality holds. Moreover, we
establish the relations between \(M_1/n\) and \(M_2/m\) in case of general graphs.
