Mathematical results on harmonic polynomials

Document Type : Original paper

Authors

1 Department of Mathematics and Statistics, Florida International University

2 Departamento de Matemáticas, Universidad Nacional de Nordeste

3 Departamento de Matemáticas. Universidad Carlos III de Madrd.

4 Departamento de Matemáticas. Universidad Autónma de Guerrero.

5 Universidad Autónoma de Guerrero

Abstract

The harmonic polynomial was defined in order to understand better the harmonic topological index. Here, we obtain several properties of this polynomial, and we prove that several properties of a graph can be deduced from its harmonic polynomial. Also, we prove that graphs with the same harmonic polynomial share many properties although they are not necessarily isomorphic.

Keywords

Main Subjects


[1] B. Borovićanin and B. Furtula, On extremal Zagreb indices of trees with given domination number, Appl. Math. Comput. 279 (2016), 208–218.
https://doi.org/10.1016/j.amc.2016.01.017
[2] K.C. Das, On comparing Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 63 (2010), no. 2, 433–440.
[3] H. Deng, S. Balachandran, S.K. Ayyaswamy, and Y.B. Venkatakrishnan, On the harmonic index and the chromatic number of a graph, Discrete Appl. Math. 161 (2013), no. 16-17, 2740–2744.
https://doi.org/10.1016/j.dam.2013.04.003
[4] Z. Du, B. Zhou, and N. Trinajstić, On the general sum-connectivity index of trees, Appl. Math. Lett. 24 (2011), no. 3, 402–405.
https://doi.org/10.1016/j.aml.2010.10.038
[5] M. Eliasi, A. Iranmanesh, and I. Gutman, Multiplicative versions of first Zagreb index, MATCH Commun. Math. Comput. Chem. 68 (2012), no. 1, 217–230.
[6] S. Fajtlowicz, On conjectures of Graffiti-II, Congr. Numer. 60 (1987), 187–197.
[7] G. Fath-Tabar, Zagreb polynomial and Pi indices of some nano structures., Digest J. Nanomat. Biostr. 4 (2009), no. 1, 189–191.
[8] O. Favaron, M. Mahéo, and J.F. Saclé, Some eigenvalue properties in graphs (conjectures of Graffiti–II), Discrete Math. 111 (1993), no. 1-3, 197–220.
http://doi.org/10.1016/0012–365X(93)90156–N
[9] B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015), no. 4, 1184–1190.
https://doi.org/10.1007/s10910-015-0480-z
[10] B. Furtula, I. Gutman, and S. Ediz, On difference of Zagreb indices, Discrete Appl. Math. 178 (2014), 83–88.
http://doi.org/10.1016/j.dam.2014.06.011
[11] I. Gutman and B. Furtula, Recent Results in the Theory of Randić Index, Univ. Kragujevac, Kragujevac, 2008.
[12] I. Gutman, B. Furtula, E. Milovanović, and I.Z. Milovanović, Bounds in Chemical Graph Theory-Mainstreams, Mathematical Chemistry Monograph no. 19, Univ. Kragujevac, Kragujevac (Serbia), 2017.
[13] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total $\varphi$-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538.
https://doi.org/10.1016/0009-2614(72)85099-1
[14] J.C. Hernández-Gómez, J.A. Méndez-Bermúdez, J.M. Rodríguez, and J.M. Sigarreta, Harmonic index and harmonic polynomial on graph operations, Symmetry 10 (2018), no. 10, Article ID: 456.
https://doi.org/10.3390/sym10100456
[15] A. Ilić, Note on the harmonic index of a graph, Ars Combin. 128 (2016), 295–299.
[16] M.A. Iranmanesh and M. Saheli, On the harmonic index and harmonic polynomial of caterpillars with diameter four, Iranian J. Math. Chem. 6 (2015), no. 1, 41–49.
https://doi.org/10.22052/ijmc.2015.9044
[17] X. Li, I. Gutman, and M. Randić, Mathematical aspects of Randić-type molecular structure descriptors, Univ. Kragujevac, Kragujevac, 2006.
[18] X. Li and Y. Shi, A survey on the Randić index, MATCH Commun. Math. Comput. Chem. 59 (2008), no. 1, 127–156.
[19] M. Liu, A simple approach to order the first Zagreb indices of connected graphs, MATCH Commun. Math. Comput. Chem. 63 (2010), no. 2, 425–432.
[20] M. Randić, Characterization of molecular branching, J. Am. Chem. Soc. 97 (1975), no. 23, 6609–6615.
http://doi.org/10.1021/ja00856a001
[21] J.A. Rodríguez and J.M. Sigarreta, On the Randić index and conditional parameters of a graph, MATCH Commun. Math. Comput. Chem. 54 (2005), no. 2, 403–416.
[22] J.M. Rodríguez and J.M. Sigarreta, New results on the harmonic index and its generalizations, MATCH Commun. Math. Comput. Chem. 78 (2017), no. 2, 387–404.
[23] J.A. Rodríguez-Velázquez and J. Tomás-Andreu, On the Randić index of polymeric networks modelled by generalized Sierpinski graphs, MATCH Commun. Math. Comput. Chem. 74 (2015), no. 1, 145–160.
[24] Y. Shi, M. Dehmer, W. Li, and I. Gutman (eds.), Graph polynomials, series: Discrete mathematics and its applications, Chapman and Hall/CRC, Taylor and Francis Group, Boca Raton, Florida, U.S.A, 2017.
[25] T. Vetrík and M. Abas, Multiplicative Zagreb indices of trees with given domination number, Commun. Comb. Optim. 9 (2024), no. 1, 89–99.
https://doi.org/10.22049/cco.2022.27972.1409
[26] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947), no. 1, 17–20.
https://doi.org/10.1021/ja01193a005
[27] R. Wu, Z. Tang, and H. Deng, A lower bound for the harmonic index of a graph with minimum degree at least two, Filomat 27 (2013), no. 1, 51–55.
https://doi.org/10.2298/FIL1301051W
[28] X. Xu, Relationships between harmonic index and other topological indices, Appl. Math. Sci. 6 (2012), no. 41, 2013–2018.
[29] S. Zafar, R. Nazir, M.S. Sardar, and Z. Zahid, Edge version of harmonic index and harmonic polynomial of some classes of graphs., J. Appl. Math. Inform. 34 (2016), no. 5-6, 479–486.
http://doi.org/10.14317/jami.2016.479
[30] L. Zhong and K. Xu, Inequalities between vertex-degree-based topological indices, MATCH Commun. Math. Comput. Chem. 71 (2014), no. 3, 627–642.
[31] B. Zhou and N. Trinajstić, On general sum-connectivity index, J. Math. Chem. 47 (2010), 210–218.
https://doi.org/10.1007/s10910-009-9542-4
[32] Z. Zhu and H. Lu, On the general sum-connectivity index of tricyclic graphs, J. Appl. Math. Comput. 51 (2016), 177–188.
http://doi.org/10.1007/s12190-015-0898-2