# Extremal Bipartite Graphs with Given Parameters on the Resistance–Harary Index

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## Abstract

**:**

## 1. Introduction

## 2. General Connected Bipartite Graphs

**Lemma**

**1**

**Lemma**

**2**

**Corollary**

**1.**

**Lemma**

**3**

**Theorem**

**1.**

**Proof.**

**Claim 1.**${G}^{0}$ is a complete bipartite graph.

**Claim 2.**${G}^{0}\cong {K}_{\lfloor \frac{n}{2}\rfloor ,\lceil \frac{n}{2}\rceil}$.

**Theorem**

**2.**

**Proof.**

## 3. Bipartite Graphs Given Number of Matchings

**Lemma**

**4.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 4. Graphs with Given Vertex Bipartiteness

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Theorem**

**4.**

## 5. Bipartite Graph with a Given Cut Edges

**Lemma**

**7.**

**Proof.**

**Lemma**

**8**

**Lemma**

**9.**

**Proof.**

**Corollary**

**2.**

**Theorem**

**5.**

**Proof.**

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Todeschini, R.; Consoni, V. Handbook of Molecular Deccriptors; Wiley-VCH: New York, NY, USA, 2002. [Google Scholar]
- Plavšić, D.; Nikolić, S.; Trinajstić, N.; Mihalić, Z. On the Harary index for the characterization of chemical graphs. J. Math. Chem.
**1993**, 12, 235–250. [Google Scholar] [CrossRef] - Ivanciuc, O.; Balaban, T.S.; Balaban, A.T. Reciprocal distance matrix related local vertex invariants and topological indices. J. Math. Chem.
**1993**, 12, 309–318. [Google Scholar] [CrossRef] - Das, K.C.; Zhou, B.; Trinajstić, N. Bounds on Harary index. J. Math. Chem.
**2009**, 46, 1369–1376. [Google Scholar] [CrossRef] - Feng, L.; Ilić, A. Zagreb, Harary and hyper-Wiener indices of graphs with a given matching number. Appl. Math. Lett.
**2010**, 23, 943–948. [Google Scholar] [CrossRef] [Green Version] - lučić, B.; Miličević, A.; Nikolić, S.; Trinajstić, N. Harary index-twelve years later. Croat. Chem. Acta
**2002**, 75, 847–868. [Google Scholar] - Zhou, B.; Cai, X.; Trinajstić, N. On the Harary index. J. Math. Chem.
**2008**, 44, 611–618. [Google Scholar] [CrossRef] - Klein, D.J.; Randić, M. Resistance distance. J. Math. Chem.
**1993**, 12, 81–95. [Google Scholar] [CrossRef] - Klein, D.J. Graph geometry, graph metrices and Wiener. MATCH Commun. Math. Comput. Chem.
**1997**, 335, 7–27. [Google Scholar] - Klein, D.J.; Zhu, H.Y. Distance and volumina for graphs. J. Math. Chem.
**1998**, 23, 179–195. [Google Scholar] [CrossRef] - Klein, D.J.; IVanciuc, O. Graph cyclicity excess conductance and resistance deficit. J. Math. Chem.
**2001**, 30, 271–287. [Google Scholar] [CrossRef] - Yang, Y. On a new cyclicity measure of graphs:The global cyclicity index. Discrete Appl. Math.
**2014**, 172, 88–97. [Google Scholar] [CrossRef] - Chen, S.; Guo, Z.; Zeng, T.; Yang, L. On the Resistance–Harary index of unicyclic graphs. MATCH Commun. Math. Comput. Chem.
**2017**, 28, 189–198. [Google Scholar] - Wang, H.; Hua, H.; Zhang, L.; Wen, S. On the Resistance–Harary index of graphs with a given number of cut edges. J. Chem.
**2017**, 12, 1–7. [Google Scholar] - Bondy, J.A.; Murty, U.S.R. Graph Theory with Applications; Macmillan London and Elsevier: New York, NY, USA, 1976. [Google Scholar]
- Klein, D.J. Resistance-distance sum rules. Croat. Chem. Acta
**2002**, 75, 633–649. [Google Scholar] - Fallat, S.; Fan, Y.Z. Bipatiteness and the least eigenvalue of signless Laplacian of graphs. Linear Algebra Appl.
**2012**, 436, 3254–3267. [Google Scholar] [CrossRef]

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Wang, H.; Yin, P.
Extremal Bipartite Graphs with Given Parameters on the Resistance–Harary Index. *Symmetry* **2019**, *11*, 615.
https://doi.org/10.3390/sym11050615

**AMA Style**

Wang H, Yin P.
Extremal Bipartite Graphs with Given Parameters on the Resistance–Harary Index. *Symmetry*. 2019; 11(5):615.
https://doi.org/10.3390/sym11050615

**Chicago/Turabian Style**

Wang, Hongzhuan, and Piaoyang Yin.
2019. "Extremal Bipartite Graphs with Given Parameters on the Resistance–Harary Index" *Symmetry* 11, no. 5: 615.
https://doi.org/10.3390/sym11050615