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Title:
More on the Zagreb indices inequality
Authors:
Imran Nadeem, Saba Siddique
doi:
Volume
87
Issue
1
Year
2022
Pages
115-123
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.

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