General Distance Energies and General Distance Estrada Index of Random Graphs
Applied and Computational Mathematics
Volume 7, Issue 3, June 2018, Pages: 173-179
Received: Aug. 9, 2018; Published: Aug. 13, 2018
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Author
Nan Gao, College of Science, Xi'an Shiyou University, Xi'an, China
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Abstract
In 2000s, Gutman and Güngör introduced the concept of distance energy and the distance Estrada index for a simple graph G respectively. Moreover, many researchers established a large number of upper and lower bounds for these two invariants. But there are only a few graphs attaining the equalities of those bounds. In this paper, however, the exact estimates to general distance energy are formulated for almost all graphs by probabilistic and algebraic approaches. The bounds to general distance Estrada index are also established for almost all graphs by probabilistic and algebraic approaches. The results of this paper generalize the results of the distance energy and distance Estrada of random graph.
Keywords
E-R Random Graph, General Distance Matrix, General Distance Energy, General Distance Estrada Index
To cite this article
Nan Gao, General Distance Energies and General Distance Estrada Index of Random Graphs, Applied and Computational Mathematics. Vol. 7, No. 3, 2018, pp. 173-179. doi: 10.11648/j.acm.20180703.24
References
[1]
F. Buckley, F. Harary, “Distance in Graphs”, Addison–Wesley, Redwood, 1990.
[2]
D. M. Cvetkovi, M. Doob, H. Sachs, “Spectra of Graphs-Theory and Application”, Academic Press, New York, 1980.
[3]
O. Ivanciuc, T. S. Balaban, A. T. Balaban, Design of topological indices. Part 4. Reciprocal distance matrix, related local vertex invariants and topological indices, J. Math. Chem. 12 (1993), pp.309-318.
[4]
I. Gutman, The energy of a graph, Ber. Math. Statist. Sekt. Forschungsz. Graz, 103 (1978), 1-22.
[5]
K. Yates, Hückel Molecular Orbital Theory, Academic Press, New York, 1978.
[6]
X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012.
[7]
G. Indulal, I. Gutman, A. Vijaykumar, On the distance energy of a graph, MATCH Commun. Math. Comput. Chem., 60 (2008), pp. 461–472.
[8]
Gutman I. The Energy of a Graph: Old and New Results [M]// Algebraic Combinatorics and Applications. Springer Berlin Heidelberg, 2001, pp.196-211.
[9]
Díaz R C, Rojo O. Sharp upper bounds on the distance energies of a graph. Linear Algebra and Its Applications, 545 (2018), pp: 55-75.
[10]
So W. A shorter proof of the distance energy of complete multipartite graphs. Special Matrices, 5 (2017), pp:61-63.
[11]
Andjelic M, Koledin T, Stanic Z. Distance spectrum and energy of graphs with small diameter. Applicable Analysis & Discrete Mathematics, 11 (2017), pp: 108-122.
[12]
G. Indulal, Sharp bounds on the distance spectral radius and the distance energy of graphs, Linear Algebra Appl., 430 (2009), pp. 106–113.
[13]
E. Estrada, Characterization of 3D molecular structure, Chem. Phys. Lett., 319 (2000), pp. 713–718.
[14]
E. Estrada, Characterization of the folding degree of proteins, Bioinformatics 18 (2002), pp: 697–704.
[15]
E. Estrada, Characterization of the amino acid contribution to the folding degree of proteins, Proteins 54 (2004), pp: 727–737.
[16]
E. Estrada, J. A. Rodríguez-Velázquez, M. Randić, Atomic branching in molecules, Int. J. Quantum Chem. 106 (2006), pp: 823–832.
[17]
E. Estrada, J. A. Rodríguez-Velázquez, Subgraph centrality in complex networks, Phys. Rev. E71 (2005) 056103-1-9.
[18]
E. Estrada, J. A. Rodríguez-Velázquez, Spectral measures of bipartivity in complex networks, Phys. Rev. E72 (2005) 046105-1-6.
[19]
R. Carbó–Dorca, Smooth fuction topological structure descriptors based on graph-spectra, J. Math. Chem. 44 (2008), pp: 373–378.
[20]
A. D. Güngör, Ş. B. Bozkurt, On the distance Estrada index of graphs, Hacettepe J. Math. Stat., 38 (2009), pp.277–283.
[21]
Ş. B. Bozkurt, D. Bozkurt, Bounds for the distance Estrada index of graphs. AIP Conf. Proc., 1648 (2015), pp. 1351–1359.
[22]
Güngör A D, Sinan A. On the Harary energy and Harary Estrada index of a graph. MATCH Commun. Math. Comput. Chem., 64 (2010), pp: 270-285.
[23]
P. Erdős, A. Rényi, On random graphs I, Publ. Math. Debrecen., 6 (1959), pp. 290–297.
[24]
W. Du, X. Li, Y. Li, The energy of random graphs, Linear Algebra Appl., 435 (2009), pp. 2334–2346.
[25]
Z. Chen, Y. Fan, W. Du, Estrada index of random graphs, MATCH Commun. Math.. Comput. Chem., 68 (2012), pp. 825–834.
[26]
W. Du, X. Li, Y. Li, Various energies of random graphs, MATCH Commun. Math. Comput. Chem., 64 (2010), pp.251–260.
[27]
Y. Shang, Distance Estrada index of random graphs, Linear Multilinear Algebra, 63 (2015), pp. 466–471.
[28]
B. Bollobás, Random Graphs, Cambridge Univ. Press, Cambridge, 2001.
[29]
K. Fan, Maximum properties and inequalities for the eigenvalues of completely con-tinuous operators, Proc. Natl. Acad. Sci. USA 37 (1951), pp. 760–766.
[30]
H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Dif-ferentialgleichungen, Math. Ann., 71 (2010), pp. 441–479.
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