Geodesic
The Minimum Harmonic Index for Unicyclic Graphs with Given Diameter
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 2, pp. 429-442.

Voir la notice de l'article provenant de la source Library of Science

The harmonic index of a graph G is defined as the sum of the weights 2d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, we present the minimum harmonic index for unicyclic graphs with given diameter and characterize the corresponding extremal graphs. This answers an unsolved problem of Zhu and Chang [26].
Keywords: harmonic index, unicyclic graphs, diameter 
@article{DMGT_2018_38_2_a6,
     author = {Zhong, Lingping},
     title = {The {Minimum} {Harmonic} {Index} for {Unicyclic} {Graphs} with {Given} {Diameter}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {429--442},
     publisher = {mathdoc},
     volume = {38},
     number = {2},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2018_38_2_a6/}
}
TY  - JOUR
AU  - Zhong, Lingping
TI  - The Minimum Harmonic Index for Unicyclic Graphs with Given Diameter
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2018
SP  - 429
EP  - 442
VL  - 38
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2018_38_2_a6/
LA  - en
ID  - DMGT_2018_38_2_a6
ER  - 
%0 Journal Article
%A Zhong, Lingping
%T The Minimum Harmonic Index for Unicyclic Graphs with Given Diameter
%J Discussiones Mathematicae. Graph Theory
%D 2018
%P 429-442
%V 38
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2018_38_2_a6/
%G en
%F DMGT_2018_38_2_a6
Zhong, Lingping. The Minimum Harmonic Index for Unicyclic Graphs with Given Diameter. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 2, pp. 429-442. http://geodesic.mathdoc.fr/item/DMGT_2018_38_2_a6/

[1] C. Betancur, R. Cruz and J. Rada, Vertex-degree-based topological indices over star-like trees, Discrete Appl. Math. 185 (2015) 18–25. doi:10.1016/j.dam.2014.12.021

[2] H. Deng, S. Balachandran and S.K. Ayyaswamy, On two conjectures of Randić index and the largest signless Laplacian eigenvalue of graphs, J. Math. Anal. Appl. 411 (2014) 196–200. doi:10.1016/j.jmaa.2013.09.014

[3] H. Deng, S. Balachandran, S.K. Ayyaswamy and Y.B. Venkatakrishnan, On the harmonic index and the chromatic number of a graph, Discrete Appl. Math. 161 (2013) 2740–2744. doi:10.1016/j.dam.2013.04.003

[4] S. Fajtlowicz, On conjectures of Graffiti— II, Congr. Numer. 60 (1987) 187–197.

[5] Q. Fan, S. Li and Q. Zhao, Extremal values on the harmonic number of trees, Int. J. Comput. Math. 92 (2015) 2036–2050. doi:10.1080/00207160.2014.965696

[6] O. Favaron, M. Mahéo and J.-F. Saclé, Some eigenvalue properties in graphs ( conjectures of Graffiti— II), Discrete Math. 111 (1993) 197–220. doi:10.1016/0012-365X(93)90156-N

[7] B. Furtula, I. Gutman and M. Dehmer, On structure-sensitivity of degree-based topological indices, Appl. Math. Comput. 219 (2013) 8973–8978. doi:10.1016/j.amc.2013.03.072

[8] I. Gutman and B. Furtula (Eds.), Recent Results in the Theory of Randić Index (University of Kragujevac, Kragujevac, 2008).

[9] I. Gutman and J. Tošović, Testing the quality of molecular structure descriptors. Vertex-degree-based topological indices, J. Serb. Chem. Soc. 78 (2013) 805–810. doi:10.2298/JSC121002134G

[10] S. He and S. Li, On the signless Laplacian index of unicyclic graphs with fixed diameter, Linear Algebra Appl. 436 (2012) 252–261. doi:10.1016/j.laa.2011.07.002

[11] J.A. Jerline and L.B. Michaelraj, On a conjecture of harmonic index and diameter of graphs, Kragujevac J. Math. 40 (2016) 73–78. doi:10.5937/KgJMath1601073J

[12] X. Li and I. Gutman, Mathematical Aspects of Randić-Type Molecular Structure Descriptors (University of Kragujevac, Kragujevac, 2006).

[13] X. Li and Y. Shi, A survey on the Randić index, MATCH Commun. Math. Comput. Chem. 59 (2008) 127–156.

[14] M.A. Iranmanesh and M. Saheli, On the harmonic index and harmonic polynomial of caterpillars with diameter four, Iran. J. Math. Chem. 6 (2015) 41–49.

[15] J. Rada and R. Cruz, Vertex-degree-based topological indices over graphs, MATCH Commun. Math. Comput. Chem. 72 (2014) 603–616.

[16] M. Randić, On characterization of molecular branching, J. Amer. Chem. Soc. 97 (1975) 6609–6615. doi:10.1021/ja00856a001

[17] B.S. Shetty, V. Lokesha and P.S. Ranjini, On the harmonic index of graph operations, Trans. Comb. 4 (2015) 5–14.

[18] M. Song and X.-F. Pan, On the Randić index of unicyclic graphs with fixed diameter, MATCH Commun. Math. Comput. Chem. 60 (2008) 523–538.

[19] R. Wu, Z. Tang and H. Deng, A lower bound for the harmonic index of a graph with minimum degree at least two, Filomat 27 (2013) 51–55. doi:10.2298/FIL1301051W

[20] K. Xu, The smallest Hosoya index of unicyclic graphs with given diameter, Math. Commun. 17 (2012) 221–239.

[21] L. Zhong, The harmonic index for graphs, Appl. Math. Lett. 25 (2012) 561–566. doi:10.1016/j.aml.2011.09.059

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

[23] L. Zhong and Q. Cui, The harmonic index for unicyclic graphs with given girth, Filomat 29 (2015) 673–686. doi:10.2298/FIL1504673Z

[24] L. Zhong and K. Xu, The harmonic index for bicyclic graphs, Util. Math. 90 (2013) 23–32.

[25] B. Zhou and N. Trinajstić, On general sum-connectivity index, J. Math. Chem. 47 (2010) 210–218. doi:10.1007/s10910-009-9542-4

[26] Y. Zhu and R. Chang, Minimum harmonic indices of trees and unicyclic graphs with given number of pendant vertices and diameter, Util. Math. 93 (2014) 365–374.