On the Harmonic Index and the Signless Laplacian Spectral Radius of Graphs

Authors: H. DENG, T. VETRíK AND S. BALACHANDRAN
DOI: 10.46793/KgJMat2102.299D
Abstract:
The harmonic index of a conected graph G is defined as H(G) = ∑ uv∈E(G)


Keywords:
Harmonic index, spectral radius, eigenvalue, signless Laplacian matrix.
References:
[1] M. Aouchiche, P. Hansen and M. Zheng, Variable neighborhood search for extremal graphs. 18. Conjectures and results about Randić index, MATCH Commun. Math. Comput. Chem. 56 (2006), 541–550.
[2] M. Aouchiche, P. Hansen and M. Zheng, Variable neighborhood search for extremal graphs. 19. Further conjectures and results about the Randić index, MATCH Commun. Math. Comput. Chem. 58 (2007), 83–102.
[3] B. Bollobás and P. Erds, Graphs of extremal weights, Ars
Comb. 50 (1998), 225–233.
[4] R. A. Brualdi and E. S. Solheid, On the spectral radius of connected graphs, Publ. Inst. Math. (Beograd) 39 (1984), 45–54.
[5] 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.
[6] 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.
[7] H. Deng, S. Balachandran, S. K. Ayyaswamy and Y. B. Venkatakrishnan, On harmonic indices of trees, unicyclic graphs and bicyclic graphs, Ars Comb. 130 (2017), 239–248.
[8] S. Fajtlowicz, On conjectures of Graffiti II, in: Combinatorics, graph theory, and computing, Proceedings of 18th Southeast Conference in Boca Raton, Florida, Congr. Numerantium 60, 1987, 189–197.
[9] O. Favaron, M. Mahéo and J.-F. Saclé, Some eigenvalue properties in graphs (conjectures of Graffiti - II), Discrete Math. 111 (1993), 197–220.
[10] L. Feng and G. Yu, On three conjectures involving the signless Laplacian spectral radius of graphs, Publ. Inst. Math. (N.S.) 85 (2009), 35–38.
[11] P. Hansen and C. Lucas, Bounds and conjectures for the signless Laplacian index of graphs, Linear Algebra Appl. 432 (2010), 3319–3336.
[12] P. Hansen and D. Vukicević, Variable neighborhood search for extremal graphs. 23. On the Randić index and the chromatic number, Discrete Math. 309 (2009), 4228–4234.
[13] Y. Hong, A bound on the spectral radius of graphs, Linear Algebra Appl. 108 (1988), 135–139.
[14] Y. Hu and X. Zhou, On the harmonic index of the unicyclic and bicyclic graphs, WSEAS Transactions on Mathematics 12 (2013), 716–726.
[15] B. Ning and X. Peng, The Randić index and signless Laplacian spectral radius of graphs, Discrete Math. 342(3) (2019), 643–653.
[16] M. Randić, On characterization of molecular branching, J. Amer. Chem. Soc. 97 (1975), 6609–6615.
[17] L. Zhong, The harmonic index on unicyclic graphs, Ars Combin. 104 (2012), 261–269.