The harmonic index of subdivision graphs

Document Type: Research Paper

Author

Golestan University

Abstract

‎The harmonic index of a graph $G$ is defined as the sum of the weights‎ ‎$\frac{2}{\deg_G(u)+\deg_G(v)}$ of all edges $uv$‎ ‎of $G$‎, ‎where $\deg_G(u)$ denotes the degree of a vertex $u$ in $G$‎. ‎In this paper‎, ‎we study the harmonic index of subdivision graphs‎, ‎$t$-subdivision graphs and also‎, ‎$S$-sum and $S_t$-sum of graphs‎.

Keywords

Main Subjects


[1] A. Astaneh-Asl and G. H. Fath-Tabar, Computing the rst and third Zagreb p olynomials of Cartesian pro duct of graphs, Iranian J. Math. Chem., 2 no. 2 (2011) 73-78.

[2] P. S. Bullen, A dictionary of inequalities, Addison-Wesley Longman, 1998.

[3] H. Deng, S. Balachandran, S. K. Ayyaswamy and Y. B. Venkatakrishnan, On the harmonic index and the chromatic numb er of a graph, Discrete Appl. Math., 161 (2013) 2740-2744.

[4] M. Eliasi and B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Appl. Math., 157 (2009) 794-803.

[5] S. Fa jtlowicz, On conjectures of graffiti I I, Congr. Numer., 60 (1987) 189-197.

[6] G. H. Fath-Tabar, Old and new Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem., 65 (2011) 79-84.

[7] A. Ilic, M. Ilic and B. Liu, On the upp er b ounds for the rst Zagreb index, Kragujevac J. Math., 35 (2011) 173-182.

[8] J. Li, J. B. Lv and Y. Liu, The harmonic index of some graphs, Bul l. Malays. Math. Sci. Soc., 39 (2016) 331-340.

[9] J. B. Lv and J. Li, On the harmonic index and the matching numb ers of trees, Ars Combin., 116 (2014) 407-416.

[10] J. B. Lv, J. Li and W. C. Shiu, The harmonic index of unicyclic graphs with given matching number, Kragujevac J. Math., 38 (2014) 173-182.

[11] S. Nikolic, G. Kovacevic, A. Milicevic and N. Trina jstic, The Zagreb indices 30 years after, Croat. Chem. Acta, 76 (2003) 113-124.

[12] B. N. Onagh, The harmonic index of pro duct graphs, Math. Sci., (2017) 1-7.

[13] B. N. Onagh, The harmonic index of edge-semitotal graphs, total graphs and related sums, Kragujevac J. Math., to app ear.

[14] B. N. Onagh, The harmonic index for R-sum of graphs, submitted.

[15] B. S. Shwetha, V. Lokesha and P. S. Ranjini, On the harmonic index of graph op erations, Trans. Comb. , 4 no. 4 (2015) 5-14.

[16] R. Wu, A. Tang and H. Deng, A lower b ound for the harmonic index of a graph with minimum degree at least two, Filomat, 27 (2013) 51-55.

[17] X. Xu, Relationships b etween harmonic index and other top ological indices, Appl. Math. Sci., 6 (2012) 2013-2018.

[18] L. Zhong, The harmonic index for graphs, Appl. Math. Lett., 25 (2012) 561-566.

[19] L. Zhong, The harmonic index on unicyclic graphs, Ars Combin., 104 (2012) 261-269.

[20] L. Zhong, On the harmonic index and the girth for graphs, Rom. J. Inf. Sci. Tech., 16 no. 4 (2013) 253-260.

[21] L. Zhong and K. Xu, The harmonic index for bicyclic graphs, Utilitas Math., 90 (2013) 23-32.

[22] L. Zhong and K. Xu, Inequalities b etween vertex-degree-based top ological indices, MATCH Commun. Math. Com-put. Chem., 71 (2014) 627-642.