On Harmonic Index and Diameter of Graphs


The harmonic index of a graph G is defined as , where d(u) denotes the degree of a vertex u in G. It has been found that the harmonic index correlates well with the Randi c' index and with the π-electronic energy of benzenoid hydrocarbons. In this work, we give several relations between the harmonic index and diameter of graphs.

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Liu, J. (2013) On Harmonic Index and Diameter of Graphs. Journal of Applied Mathematics and Physics, 1, 5-6. doi: 10.4236/jamp.2013.13002.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] J. A. Bondy and U. S. R. Murty, “Graph Theory,” Springer, Berlin, 2008.
[2] X. Li, I. Gutman, “Mathematical Aspects of Randic-Type Molecular Structure Descriptors,” Mathematical Chemistry Monographs No.1, University of Kragujevac, 2006.
[3] X. Li and Y. T. Shi, “A Survey on the Randic Index,” Communications in Mathematical and in Computer Chemistry, Vol. 59, No. 1, 2008, pp. 127-156.
[4] B. Lucic, N. Trinajstic and B. Zhou, “Comparison between the Sum-Connectivity Index and ProductConnectivity Index for Benzenoid Hydrocarbons,” Chemical Physics Letters, Vol. 475, No. 1-3, 2009, pp. 146-148. doi:10.1016/j.cplett.2009.05.022
[5] B. Lucic, S. Nikolic, N. Trinajstic, B. Zhou and S. I. Turk, “Sum-Connectivity Index,” In: I. Gutman and B. Furtula, Ed., Novel Molecular Structure Descriptors-Theory and Applications I, University of Kragujevac, Kragujevac, 2010, pp. 101-136.
[6] O. Favaron, M. Mahó and J. F. Saclé, “Some Eigenvalue Properties in Graphs (Conjectures of Graffiti-II),” Discrete Mathematics, Vol. 111 No. 1-3, 1993, pp. 197-220. doi:10.1016/0012-365X(93)90156-N
[7] L. Zhong, “The Harmonic Index for Graphs,” Applied Mathematics Letters, Vol. 25, No. 3, 2012, pp. 561-566. doi:10.1016/j.aml.2011.09.059
[8] R. Wu, Z. Tang and H. Deng, “A Lower Bound for the Harmonic Index of a Graph with Minimum Degree at Least Two,” Filomat, Vol. 27, No. 1, 2013, pp. 51-55.

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