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Some tight bounds for the harmonic index and the variation of the Randić index of graphs. (English) Zbl 1414.05083

Summary: The harmonic index \(H(G)\) and the variation of the Randić index \(R^\prime(G)\) of a graph \(G\) are defined as the sum of the weights \(\frac{2}{d_u+d_v}\) and \(\frac{1}{\max\{d_u, d_v\}}\) over all the edges \(uv\) of \(G\), respectively, where \(d_u\) denotes the degree of a vertex \(u\) in \(G\). In this paper, we give some tight bounds for the harmonic index (or the variation of the Randić index, respectively) of \(G\) in terms of its maximum and minimum degree mean rates over its edges, where the degree mean rate is \(\gamma_H(e)=\frac{2d_ud_v}{(d_u+d_v)^2}\) (or \(\gamma_{R^\prime}(e)=\frac{\min\{d_ud_v\}}{d_u+d_v}\), respectively) of an edge \(e=uv\in E\). We use the same method given by C. Dalfó [“On the Randić index of graphs”, Discrete Math. (to appear)], which proved the conjecture by Fajtlowicz that states that the average distance is bounded above by the Randić index, for graphs with order \(n\) large enough when \(\delta\) is greater than (approximately) \(\varDelta^{\frac13}\), where \(\varDelta\) is the maximum degree.

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05C07 Vertex degrees

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