Research PaperNo Access

On the computation of extremal trees of Harmonic index with given edge-vertex domination number

B. Senthilkumar

https://orcid.org/0000-0001-8121-7233

School of Arts Sciences Humanities and Education, SASTRA Deemed to be University, Thanjavur 613 401, India

E-mail Address: senthilsubramanyan@gmail.com

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H. Naresh Kumar

School of Arts Sciences Humanities and Education, SASTRA Deemed to be University, Thanjavur 613 401, India

E-mail Address: nareshhari1403@gmail.com

Corresponding author.

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Y. B. Venkatakrishnan

School of Arts Sciences Humanities and Education, SASTRA Deemed to be University, Thanjavur 613 401, India

E-mail Address: ybvenkatakrishnan2@gmail.com

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, and

S. P. Raja

School of Computer Science and Engineering, Vellore Institute of Technology, Vellore 632 014, Tamilnadu, India

E-mail Address: avemariaraja@gmail.com

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https://doi.org/10.1142/S0219691323500145Cited by:1 (Source: Crossref)

Abstract

Let $v_{1}, v_{2}$ be vertices of a graph $G$ with degree of the vertices being $d (v_{1})$ and $d (v_{2}),$ respectively. First, let us define the weight of the edge $v_{1} v_{2}$ as twice the value of $\frac{1}{d (v_{1}) + d (v_{2})}$ in $G$ . Let us define $H (G)$ , the harmonic index of the graph $G$ , as the sum obtained by adding the weight assigned to every edge of $G$ . In this paper, for the class of trees, we shall obtain an upper bound for the harmonic index $H (G)$ in terms of the edge-vertex domination number and the order of $G$ . Also, we shall ascertain that the equality is true by characterizing the collection of all extremal trees attaining this bound.

Keywords:

AMSC: 05C07, 05C35, 05C92

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