New Bounds for the Harary Energy and Harary Estrada index of Graphs

Abstract

The Harary index is defined as the sum of reciprocal distances between all

pairs of vertices in a nontrivial connected graph. In this paper, we establish

bounds for the Harary energy and Harary Estrada index in terms of graph

invariants such as the number of vertices, the number and spectral radius.

Keywords

• Eigenvalue of graph,Harary Energy,Spectral radius

References

1. [1] R. Binthiya, B. Sarasija, A note on strongly quotient graphs with Harary energyand Harary Estrada index, App. Math. E-Notes . 14, (2014), 97-106.
2. [2] Z. Cui, B. Liu, On Harary matrix, Harary index and Harary energy, MATCHCommun. Math. Comput. Chem. 68, (2012), 815-823.
3. [3] D. Cvetkovic, P. Rowlinson, S. Simic, An Introduction to the Theory of GraphSpectra, Cambridge Univ. Press, Cambridge, 2010.
4. [4] K. C. Das, Maximum eigenvalues of the reciprocal distance matrix,J. Math.Chem. 47, (2010), 21-28.
5. [5] J. A. De la Pe~na, I. Gutman, J. Rada, Estimating the Estrada Index, Lin.Algebra Appl. 427, (2007) 70-76.
6. [6] H. Deng, S. Radenkovic, I. Gutman, The Estrada Index, in: D. Cvetkovic, I.Gutman (Eds.), Applications of Graph Spectra, Math. Inst., Belgrade, 2009,123-140.
7. [7] M. V. Diudea, O. Ivanciuc, S. Nikolic, N. Trinajstic, Matrices of reciprocaldistance, polynomials and derived numbers, MATCH Commun. Math. Comput.Chem. 35, (1997), 41-64.
8. [8] A. D. Gungor, A. S. Cevik, On the Harary energy and Harary Estrada indexof a graph, MATCH Commun. Math. Comput. Chem. 64, (2010), 280-296.

9. [9] I. Gutman, M. Milun, N. Trinajstic, Comment on the paper: Properties of thelatent roots of a matrix. Estimation of -electron energies by B. J. McClelland,J. Chem. Phys. 59 (1973), 2772-2774.
10. [10] I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry,Springer, Berlin, 1986.
11. [11] I. Gutman, The energy of a graph: old and new results, in: A. Betten, A.Kohnert, R. Laue and A. Wassermann (Eds.), Algebraic Combinatorics andApplications, Springer-Verlag, Berlin, (2001), 196-211.
12. [12] F. Huang, X. Li, S. Wang, On graphs with maximum Harary spectral radius,arXiv:1411.6832v1 [math.CO], 25 Nov 2014.
13. [13] O. Ivanciuc, T. S. Balaban, A. T. Balaban, Design of topological indices. Part4. Reciprocal distance matrix, related local vertex invariants and topologicalindices,J. Math. Chem. 12, (1993), 309-318.
14. [14] N. Jafari. Rad, A. Jahanbani, D. A. Mojdeh, Tetracyclic Graphs with MaximalEstrada Index, Discrete Mathematics, Algorithms and Applications, 09 (2017),1750041.
15. [15] N. Jafari. Rad, A. Jahanbani, R. Hasni, Pentacyclic Graphs with MaximalEstrada Index, Ars Combin,133 (2017), 133-145.
16. [16] N. Jafari. Rad, A. Jahanbani, I. Gutman, Zagreb Energy and Zagreb EstradaIndex of Graphs,MATCH Commun. Math. Comput. Chem, 79 (2018), 371-386.
17. [17] A. Jahanbani, Upper bounds for the energy of graphs, MATCH Commun.Math. Comput. Chem. pp. 275-286.
18. [18] A. Jahanbani, Some new lower bounds for energy of graphs, Applied Mathe-matics and Computation, 296 ( 2017), 233-238.
19. [19] A. Jahanbani, Lower Bounds for the Energy of Graphs, AKCE InternationalJournal of Graphs and Combinatorics,15 (2018) 88-96.
20. [20] D. Jenezic, A. Miliccevic, S. Nikolic, N. Trinajstic, Graph Theoretical Matricesin Chemistry, Univ. Kragujevac, Kragujevac, 2007.
21. [21] X. Li, Y. Shi, I. Gutman, Graph Energy,Springer, New York, 2012.
22. [22] A. M. Mercer, P. Mercer, Cahys interlace theorem and lower bounds for thespectral radius, Internat. J. Math. and Math. Sci. 23, No. 8 (2000) 563-566.
23. [23] O. Rojo R. Soto, H. Rojo, Bounds for the spectral radius and the largestsingular value, Computers Math. Applic. 36, (1998) 41-50.
24. [24] B. Zhou, N. Trinajstic, Maximum eigenvalues of the reciprocal distance matrixand the reverse Wiener matrix,Int. J. Quantum Chem. 108, (2008), 858-864.