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On the Balaban Index of Chain Graphs

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Abstract

The Balaban index and sum-Balaban index of a connected (molecular) graph G are defined as

$$\begin{aligned} J(G)&=\frac{m}{\mu +1} \sum _{uv\in E(G)}\frac{1}{\sqrt{\sigma _{G}(u)\sigma _{G}(v)}}~ \text{ and }\\ SJ(G)&=\frac{m}{\mu +1} \sum _{uv\in E(G)}\frac{1}{\sqrt{\sigma _{G}(u)+\sigma _{G}(v)}}, \end{aligned}$$

respectively, where m is the number of edges, \(\mu \) is the cyclomatic number, \(\sigma _G(u)\) is the sum of distances between vertex u and all other vertices of G. In this paper, we establish that

$$\begin{aligned} K\left( DS(n-3,\,1)\right)>K\left( DS(n-4,\,2)\right)>\cdots >K \left( DS\left( \left\lceil \frac{n}{2}\right\rceil -1,\, \left\lfloor \frac{n}{2}\right\rfloor -1\right) \right) \end{aligned}$$

\((K=J,\,SJ),\) where \(DS(p,\,q)\) is a double star on \(n\,(=p+q+2,\,p\ge q)\) vertices. As an application, we determine the extremal graphs of the Balaban index and the sum-Balaban index in the class of chain graphs G on n vertices, where G is a tree or a unicyclic graph. Finally, we give an open problem on Balaban (sum-Balaban) index of connected chain graphs.

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Acknowledgements

The author is much grateful to two anonymous referees for their valuable comments on our paper, which have considerably improved the presentation of this paper.

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  1. Department of Mathematics, Sungkyunkwan University, Suwon, 16419, Republic of Korea

    Kinkar Chandra Das

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  1. Kinkar Chandra Das
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Correspondence to Kinkar Chandra Das.

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Communicated by Xueliang Li.

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Das, K.C. On the Balaban Index of Chain Graphs. Bull. Malays. Math. Sci. Soc. 44, 2123–2138 (2021). https://doi.org/10.1007/s40840-020-01054-y

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