Abstract
For a (molecular) graph \(G,\) the general Randić index \(R_{\alpha }(G)\) is defined as the sum of the weights \([d_{u}d_{v}]^{\alpha }\) of all edges \(uv\) of \(G,\) where \(d_{u}\) (or \(d_{v}\)) denotes the degree of a vertex \(u\) (or \(v\)) in \(G\) and \(\alpha \) is an arbitrary real number. In this paper, we give an efficient formula for computing the general Randić index of polyomino chains and characterize the extremal polyomino chains with respect to this index, which generalizes one of the main results in (Yarahmadi et al. Appl Math Lett 25:166–171, 2012).
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References
Bondy JA, Murty USR (1976) Graph Theory with Applications. Macmillan, Elsevier, London
Gutman I, Trinajstić N (1972) Graph theory and molecular orbitals, total \(\pi \)-electron energy of alternant hydrocarbons. Chem Phys Lett 17:535–538
Ivanciuc O, Ivanciuc T, Cabrol-Bass D, Balaban AT (2000) Evaluation in quantitative structure-property relationship models of structural descriptors derived from information-theory operators. J Chem Inf Comput Sci 40:631–643
Klarner DA (1997) Polyominoes. In: Goodman JE, O’Rourke J (eds) Handbook of Discrete and Computational Geometry, Chapter 12. CRC Press LLC, Boca Raton, pp. 225–242
Li X, Gutman I (2006) Mathematical Aspects of Randić-Type Molecular Structure Description. University of Kragujevac, Kragujevac
Nikolić S, Kovac̆ević G, Milic̆ević A, Trinajstić N (2003) The Zagreb indices 30 years after. Croat Chem Acta 76:113–124
Randić M (1975) On characterization of molecular branching. J Am Chem Soc 97:6609–6615
Todeschini R, Consonni V (2000) Handbook of Molecular Descriptors. Wiley-VCH, Weinheim
Xu L, Chen S (2008) The PI index of polyomino chains. Appl Math Lett 21:1101–1104
Yang J, Xia F, Chen S (2011) On the Randić index of polyomino chains. Appl Math Sci 5(5):255–260
Yang J, Xia F, Chen S (2011) On sum-connectivity index of polyomino chains. Appl Math Sci 5(6):267–271
Yarahmadi Z, Ashrafi A, Moradi S (2012) Extremal polyomino chains with respect to Zagreb indices. Appl Math Lett 25:166–171
Zhou B, Luo W (2009) A note on general Randić index. MATCH Commun Math Comput Chem 62:155–162
Zhou B, Trinajstić N (2009) On a novel connectivity index. J Math Chem 46:1252–1270
Zeng Y, Zhang F (2007) Extremal polyomino chains on k-matchings and k-independent sets. J Math Chem 42(2):125–140
Acknowledgments
The authors would like to thank the anonymous referees for their valuable comments and suggestions, which helped to improve the presentation of the paper. This work is supported by the Natural Science Funds of China (No. 11071016, 11171129 and 11001197) and by Specialized Research Fund for the Doctoral Program of Higher Education (No. 20131101110048).
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An, M., Xiong, L. Extremal polyomino chains with respect to general Randić index. J Comb Optim 31, 635–647 (2016). https://doi.org/10.1007/s10878-014-9781-6
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DOI: https://doi.org/10.1007/s10878-014-9781-6