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The minimum value of the harmonic index for a graph with the minimum degree two

Hanyuan Deng

College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P. R. China

E-mail Address: hydeng@hunnu.edu.cn

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S. Balachandran

Department of Mathematics, School of Humanities and Sciences, SASTRA Deemed University, Thanjavur, India

E-mail Address: bala_maths@rediffmail.com

Corresponding author.

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, and

S. Raja Balachandar

Department of Mathematics, School of Humanities and Sciences, SASTRA Deemed University, Thanjavur, India

E-mail Address: srbala09@gmail.com

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https://doi.org/10.1142/S1793557120500540Cited by:4 (Source: Crossref)

Abstract

For a simple and connected graph $G$ with $n$ vertices and the minimum degree two, we show that $H (G) \geq 4 + \frac{1}{n - 1} - \frac{12}{n + 1}$ by a technique based on linear programming, where $H (G)$ is the harmonic index of a graph $G$ , defined as the sum of the weights $\frac{2}{d_{u} + d_{v}}$ of all edges $u v$ of $G$ , $d_{u}$ denotes the degree of a vertex $u$ , and characterize the graph with the minimum value.

Communicated by J. Koppitz

Keywords:

AMSC: 05C07, 05C15

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