AM  Vol.4 No.8 , August 2013
On the Harmonic Index of Triangle-Free Graphs
Abstract: The harmonic index of a graph G  is defined as where d(u) denotes the degree of a vertex u in G . In this work, we give another expression for the Harmonic index. Using this expression, we give the minimum value of the harmonic index for any triangle-free graphs with order n and minimum degree δ ≥ k for k≤ n/2  and show the corresponding extremal graph is the complete graph.
Cite this paper: J. Liu, "On the Harmonic Index of Triangle-Free Graphs," Applied Mathematics, Vol. 4 No. 8, 2013, pp. 1204-1206. doi: 10.4236/am.2013.48161.

[1]   J. A. Bondy and U. S. R. Murty, “Graph Theory,” Springer, 2008. doi:10.1007/978-1-84628-970-5

[2]   X. Li and I. Gutman, “Mathematical Aspects of Randic’ Type Molecular Structure Descriptors,” Mathematical Chemistry Monographs, Vol. 1, Kragujevac, 2006.

[3]   X. Li and Y. T. Shi, “A Survey on the Randic Index,” MATCH: Communications in Mathematical and in Com puter Chemistry, Vol. 59, No. 1, 2008, pp. 127-156.

[4]   B. Lucic, S. Nikolic, N. Trinajstic, B. Zhou and S. I. Turk, “Sum-Connectivity Index,” In: I. Gutman and B. Furtula, Eds., Novel Molecular Structure Descriptors-Theory and Applications I, University of Kragujevac, Kragujevac, 2010, pp. 101-136.

[5]   B. Lucic, N. Trinajstic and B. Zhou, “Comparison be tween the Sum-Connectivity Index and Product-Con nectivity Index for Benzenoid Hydrocarbons,” Chemical Physics Letters, Vol. 475, No. 1-3, 2009, pp. 146-148. doi:10.1016/j.cplett.2009.05.022

[6]   O. Favaron, M. Mahó and J. F. Saclé, ”Some Eigenvalue Properties in Graphs (Conjectures of Graffiti-II),” Dis crete 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.