 Journal of Applied Mathematics and Physics, 2013, 1, 5-6 http://dx.doi.org/10.4236/jamp.2013.13002 Published Online August 2013 (http://www.scirp.org/journal/jamp) On Harmonic Index and Diameter of Graphs* Jianxi Liu School of Informatics, Guangdong University of Foreign Studies, Guangzhou, China Email: liujianxi2001@gmail.com, ljx@oamail.gdufs.edu.cn Received June 12, 2013; revised July 15, 2013; accepted August 5, 2013 Copyright © 2013 Jianxi Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT The harmonic index of a graph is defined as G 2uvE GHG du dv, where denotes the degree of a vertex in . It has been found that the harmonic index correlates well with the Randi index and with the -electronic energy of benzenoid hydrocarbons. In this work, we give several relations between the harmonic index and diameter of graphs. duuGcπ Keywords: Harmonic Index; Diameter 1. Introduction All graphs considered in the following will be simple. Let be a graph with vertex set and edge set . The order of graph G is the number of its vertices. The leaf of a graph is a vertex with degree one. For undefined terminology and notations we refer the reader to . For a graph G, the harmonic index GVGEGHG is defined as  2.uvE GHG du dv It has been found that the harmonic index correlates well with the Randić index [2,3] and the -electronic energy of benzenoid hydrocarbons [4,5]. Favaron et al.  considered the relation between harmonic index and the eigenvalues of graphs. Zhong  found the minimum and maximum values of the harmonic index for con- nected graphs and trees, and characterized the cor- responding extremal graphs. Recently, Wu et al.  gave the minimum value of the Harmonic index among the graphs with the minimum degree at least two. In this work, we shall give some relations between the harmonic index and diameter of graphs. π2. Main Results First, we shall give two sharp upper bounds of the relationship involving the harmonic index and diameter of connected graphs. In , Zhong gave the following result: Lemma 2.1 () Let be a graph with order , Gnthen ,2nHG where the equality holds if and only if G is a regular graph. Since the complete graph nK is a regular graph with diameter one, we have: Theorem 2.2 Let be a connected graph with order G4n diameter DT , then  1, 22HTnnHT DTDT  where equalities hold if and only if is the complete graph GnK. An edge 12xx is called local maximum if its weight 12x2dx d is maximum in its neighborhood, i.e.,  1222idx dxdx du for any edge ixu for 1, 2.i Lemma 2.3 Let 12xx be a local maximum edge in graph , then G12 0.HGHG xx *Research supported by the National Natural Science Foundation of China (No.11101097) and Foundation for Distinguished Young Tal-ents in HigherEducation of Guangdong, China (No.LYM11061). Proof. We have Copyright © 2013 SciRes. JAMP J. X. Liu 6             122112121 11221212 12212 1212 1222212222 2()11122 21011uNx xvNx xHGHG xxdx dxdx dudx dudxdxdvdxdvdx dxdx dxdx dxdx dx dxdx dxdx dxdx dx    , where denotes the vertex set adjoining to iNx ix for . 1, 2iIf 12xx is a leaf of , i.e., at least one of G12,xx has degree one, we can see that it is a local maximum edge. Thus, by Lemma 2.3, Corollary 2.4 If 12xx is a leaf in graph G, then  12 0.HGHG xx Theorem 2.5 Let T be a tree with order 4n diameter , then DT  51, 6223 1HTnHT DTDT n 1 where equalities hold if and only if T is a path . nPProof. If is a path, we have T126nHT and 1DT n. It is obvious that both equalities hold. Now we assume that T is not a path, then and there are at least three pendent vertices in 2DT nT. Assume 01DPuu u be the longest path in T. Then at least one pendent vertex, say 1, is not contained in vP. Now we start an operation on T, i.e., we continually delete pendent vertices which are not contained in P until the resulting tree is P. Assume 1 are the vertices in the order they were deleted, we have ,,kvv 11123kiiDHTHT vHTvHP  by Corollary 2.4 and  11kiiDTDT vDTvD. Thus, we have  >51515>62 6262HTDTHPDPDn n and  15 12626>= >1DnHT HPDT DPDn. This result seems true for any connected graph with order and we propose as a conjecture as follows: nConjecture 2.6 Let be a connected graph with order G4n diameter , then DG  51, 6223 1HGnHG DGDG n 1. REFERENCES  J. A. Bondy and U. S. R. Murty, “Graph Theory,” Springer, Berlin, 2008.  X. Li, I. Gutman, “Mathematical Aspects of Randić-Type Molecular Structure Descriptors,” Mathematical Chemis-try Monographs No.1, University of Kragujevac, 2006.  X. Li and Y. T. Shi, “A Survey on the Randić Index,” Communications in Mathematical and in Computer Chemistry, Vol. 59, No. 1, 2008, pp. 127-156.  B. Lučić, N. Trinajstić and B. Zhou, “Comparison be- tween the Sum-Connectivity Index and Product- Connec- tivity Index for Benzenoid Hydrocarbons,” Chemical Physics Letters, Vol. 475, No. 1-3, 2009, pp. 146-148. doi:10.1016/j.cplett.2009.05.022  B. Lučić, S. Nikolić, N. Trinajstić, B. Zhou and S. I. Turk, “Sum-Connectivity Index,” In: I. Gutman and B. 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