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Inverse degree, Randić index and harmonic index of graphs. (English) Zbl 1499.05115

Summary: Let \(G\) be a graph with vertex set \(V\) and edge set \(E\). Let \(d_i\) be the degree of the vertex \(v_i\) of \(G\). The inverse degree, Randić index, and harmonic index of \(G\) are defined as \(ID = \sum_{v_i\in V} 1/d_i, R = \sum_{v_iv_j \in E} 1/\sqrt{d_id_j}\), and \(H= \sum_{v_iv_j \in E} 2/(d_i + d_j)\), respectively. We obtain relations between \(ID\) and \(R\) as well as between \(ID\) and \(H\). Moreover, we prove that in the case of trees, \(ID > R\) and \(ID > H\).

MSC:

05C09 Graphical indices (Wiener index, Zagreb index, Randić index, etc.)
05C07 Vertex degrees
05C35 Extremal problems in graph theory

Software:

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References:

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