Higher-order Randić index and isomorphism of double starlike trees

Zhenhua Su; Zikai Tang; Hanyuan Deng; Zhenhua Su; Zikai Tang; Hanyuan Deng

doi:10.3934/math.20231596

AIMS Mathematics

2023, Volume 8, Issue 12: 31186-31197. doi: 10.3934/math.20231596

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Research article

Higher-order Randić index and isomorphism of double starlike trees

1.
School of Mathematics and Computational Sciences, Huaihua University, Huaihua, Hunan 418008, China
2.
College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China

Received: 15 September 2023 Revised: 13 November 2023 Accepted: 19 November 2023 Published: 23 November 2023
MSC : 05C05, 05C09, 05C92

For an integer $ h\geq 0 $, the $ h $th order Randić index for a simple graph $ G $ is defined as $ R^{h}(G) = \sum_{\pi}\frac{1}{\sqrt{v_1(\pi)v_2(\pi)\cdots v_{h+1}(\pi)}} $, where $ \pi $ extends over all paths of length $ h $ in $ G $ and $ v_i(\pi) $ denotes the degree of the $ i $-th vertex of the path $ \pi $. In this paper, we showed that the $ h $th order Randić index $ R^{h}(T) $ of a double starlike tree $ T $ (a tree with two vertices of degrees $ m_1, m_2 > 2 $) is completely determined by its branches of length $ \leq h $. As a consequence, we proved that the double starlike trees with equal $ h $-Randić index are isomorphic, except for some special values for $ m_1 $ and $ m_2 $.
- Randić index,
- double starlike tree,
- degree,
- isomorphism
Citation: Zhenhua Su, Zikai Tang, Hanyuan Deng. Higher-order Randić index and isomorphism of double starlike trees[J]. AIMS Mathematics, 2023, 8(12): 31186-31197. doi: 10.3934/math.20231596

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Abstract

For an integer $ h\geq 0 $, the $ h $th order Randić index for a simple graph $ G $ is defined as $ R^{h}(G) = \sum_{\pi}\frac{1}{\sqrt{v_1(\pi)v_2(\pi)\cdots v_{h+1}(\pi)}} $, where $ \pi $ extends over all paths of length $ h $ in $ G $ and $ v_i(\pi) $ denotes the degree of the $ i $-th vertex of the path $ \pi $. In this paper, we showed that the $ h $th order Randić index $ R^{h}(T) $ of a double starlike tree $ T $ (a tree with two vertices of degrees $ m_1, m_2 > 2 $) is completely determined by its branches of length $ \leq h $. As a consequence, we proved that the double starlike trees with equal $ h $-Randić index are isomorphic, except for some special values for $ m_1 $ and $ m_2 $.

References

[1]	D. Plavaic, S. Nikolic, N. Trinajstic, Z. Mihalic, On the Harary index for the characterization of chemical graphs, J. Math. Chem., 12 (1993), 235–250. https://doi.org/10.1007/BF01164638 doi: 10.1007/BF01164638
[2]	N. Ghareghani, F, Ramezani, B. Tayfeh-Rezaie, Graphs cospectral with starlike trees, Lin. Alg. Appl., 429 (2008), 2691–2701. https://doi.org/10.1016/j.laa.2008.01.001 doi: 10.1016/j.laa.2008.01.001
[3]	O. Araujo, J. A. de la Peña, The connectivity index of a weighted graph, Lin. Alg. Appl., 283 (1998), 171–177. https://doi.org/10.1016/S0024-3795(98)10096-4 doi: 10.1016/S0024-3795(98)10096-4
[4]	I. Gutman, M. Randić, Algebraic characterization of skeletal branching, Chem. Phys. Lett., 47 (1977), 15–19. https://doi.org/10.1016/0009-2614(77)85296-2 doi: 10.1016/0009-2614(77)85296-2
[5]	M. Randić, On the characterization of molecular branching, J. Am. Chem. Soc., 97 (1975), 6609–6615.
[6]	L. B. Kier, L. H. Hall, Molecular connectivity in structure-activity analysis, New York: Wiley Press, 1986.
[7]	J. Du, Y. Shao, X. Sun, The zeroth-order general Randić index of graph with a given clique number, Korean J. Math., 28 (2020), 405–419. https://doi.org/10.11568/KJM.2020.28.3.405 doi: 10.11568/KJM.2020.28.3.405
[8]	I. Gutman, Some less familiar properties of Randić index, Croatica Chemica Acta, 93 (2021), 273–278. https://doi.org/10.5562/cca3747 doi: 10.5562/cca3747
[9]	B. Furtula, A. Graovac, D, Vukicevic, Atom-bond connectivity index of trees, Discrete Appl. Math., 157 (2009), 2828–2835. https://doi.org/10.1016/j.dam.2009.03.004 doi: 10.1016/j.dam.2009.03.004
[10]	L. Volkmann, Sufficient conditions on the zeroth-order general Randić index for maximally edge-connected digraphs, Communications in Combinatorics and Optimization, 1 (2016), 1–13.
[11]	I. G. Yero, J. A. Rodríguez-Velázquez, I. Guyman, Estimating the higher-order Randić index, Chem. Phys. Lett., 489 (2010), 118–120. https://doi.org/10.1016/j.cplett.2010.02.052 doi: 10.1016/j.cplett.2010.02.052
[12]	O. Araujo, J. Rada, Randić index and lexicographic order, J. Math. Chem., 27 (2000), 201–212. https://doi.org/10.1023/A:1026424219580 doi: 10.1023/A:1026424219580
[13]	M. Randić, Representation of molecular graphs by basic graphs, J. Chem. Inf. Comput. Sci., 32 (1992), 57–69. https://doi.org/10.1021/ci00005a010 doi: 10.1021/ci00005a010
[14]	J. Rada, O. Araujo, Higher order connectivity index of starlike trees, Discrete Appl. Math., 119 (2002), 287–295. https://doi.org/10.1016/S0166-218X(01)00232-3 doi: 10.1016/S0166-218X(01)00232-3
[15]	R. Song, Q. Huang, A relation on trees and the topological indices based on subgraph, MATCH Commun. Math. Comput. Chem., 89 (2023), 343–370. https://doi.org/10.46793/match.89-2.343S doi: 10.46793/match.89-2.343S

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