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

Download PDF

**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)}, where E(G) is the edge set of G, d(u) and d(v) are the degrees of vertices u and v, respectively. The spectral radius of a square matrix M is the maximum among the absolute values of the eigenvalues of M. Let q(G) be the spectral radius of the signless Laplacian matrix Q(G) = D(G) + A(G), where D(G) is the diagonal matrix having degrees of the vertices on the main diagonal and A(G) is the (0, 1) adjacency matrix of G. The harmonic index of a graph G and the spectral radius of the matrix Q(G) have been extensively studied. We investigate the relationship between the harmonic index of a graph G and the spectral radius of the matrix Q(G). We prove that for a connected graph G with n vertices, we have

**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.