The minimum value of the harmonic index for a graph with the minimum degree two. (English) Zbl 1441.05053
Summary: For a simple and connected graph \(G\) with \(n\) vertices and the minimum degree two, we show that \(H(G)\geq4+\frac{ 1}{ n - 1}-\frac{ 1 2}{ n + 1}\) by a technique based on linear programming, where \(H(G)\) is the harmonic index of a graph \(G\), defined as the sum of the weights \(\frac{ 2}{ d_u + d_v}\) of all edges \(uv\) of \(G\), \(d_u\) denotes the degree of a vertex \(u\), and characterize the graph with the minimum value.
MSC:
| 05C09 | Graphical indices (Wiener index, Zagreb index, Randić index, etc.) |
| 05C07 | Vertex degrees |
| 05C15 | Coloring of graphs and hypergraphs |
| 05C40 | Connectivity |
| 90C05 | Linear programming |
Software:
GRAFFITIReferences:
| [1] | Dayanand, G. K. and Ahmed, Shabbir, The super magic properties of connected and disconnected graphs, J. Inform. Opt. Sci.36(3) (2015) 231-246. |
| [2] | Deng, H., Balachandran, S. and Ayyaswamy, S. K., On two conjectures of Randić index and the largest signless Laplacian eigenvalue of graphs, J. Math. Anal. Appl.411 (2014) 196-200. · Zbl 1308.05071 |
| [3] | Deng, H., Balachandran, S., Ayyaswamy, S. K. and Venkatakrishnan, Y. B., On the harmonic index and the chromatic number of a graph, Discrete Appl. Math.161 (2013) 2740-2744. · Zbl 1285.05055 |
| [4] | Deng, H., Huang, G. and Jiang, X., A unified linear-programming modeling of some topological indices, J. Comb. Optim.30 (2015) 826-837. · Zbl 1332.90324 |
| [5] | Fajtlowicz, S., On conjectures of Graffiti-II, Cong. Numer.60 (1987) 187-197. · Zbl 0713.05054 |
| [6] | Favaron, O., Mahio, M. and Saclé, J. F., Some eigenvalue properties in graphs (Conjectures of Graffiti-II), Discrete Math.111 (1993) 197-220. · Zbl 0785.05065 |
| [7] | Jayasekaran, C. and Raj, C. David, Harmonic mean labeling of disconnected graphs, J. Discrete Math. Sci. Cryptogr.19 (2016) 1-12. · Zbl 1495.05289 |
| [8] | Pavlović, L., Graphs with extremal Randić index when the minimum degree of vertices is two, Kragujevac J. Math.25 (2003) 55-63. · Zbl 1054.05059 |
| [9] | Wu, R., Tang, Z. and Deng, H., A lower bound for the harmonic index of a graph with minimum degree at least two, Filomat27 (2013) 51-55. · Zbl 1324.05030 |
| [10] | Zhao, K., Neighborhood conditions and Hamiltonian-connected graphs, J. Interdisciplinary Math.16(2-3) (2013) 137-145. |
| [11] | Zhong, L., The harmonic index for graphs, Appl. Math. Lett.25 (2012) 561-566. · Zbl 1243.05126 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
