Estimating the distance Estrada index

Yilun Shang


Suppose $G$ is a simple graph on $n$ vertices. The $D$-eigenvalues$\mu_1,\mu_2,\cdots,\mu_n$ of $G$ are the eigenvalues of itsdistance matrix. The distance Estrada index of $G$ is defined as$DEE(G)=\sum_{i=1}^ne^{\mu_i}$. In this paper, we establish newlower and upper bounds for $DEE(G)$ in terms of the Wiener index$W(G)$. We also compute the distance Estrada index for some concretegraphs including the buckminsterfullerene $C_{60}$.


Distance degree; distance matrix; Estrada index; Wiener index.

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Alaeiyan, M., Gilani, A., Mojarad, R. & Asadpour, A. (2014). Omega

and related polynomials of polyomino chains of 4 k-cycles. Kuwait

Journal of Science, 41:85-92.

Balasubramanian, K. (1995). A topological analysis of the C60

buckminsterfullerene and C70 based on distance matrices. Chemical

Physics Letters, 239:117-123.

Bapat, R.B. & Sivasubramanian, S. (2013). Product distance matrix

of a graph and squared distance matrix of a tree. Applicable Analysis

and Discrete Mathematics, 7:285-301.

Bozkurt, Ş.B., Adiga, C. & Bozkurt, D. (2013). Bounds on the

distance energy and the distance Estrada index of strongly quotient

graphs. Journal of Applied Mathematics, 2013:681019.

Bozkurt, Ş.B. & Bozkurt, D. (2012). Bounds for the distance Estrada

index of graphs. Available at arXiv:1205.1189.

Cvetković, D.M., Doob, M. & Sachs, H. (1995). Spectra of Graphs:

Theory and Application. Johann Ambrosius Bart Verlag, Heidelberg.

Das, K.C. & Lee, S.G. (2009). On the Estrada index conjecture. Linear

Algebra and its Applications, 431:1351- 1359.

de la Peña, J.A., Gutman, I. & Rada, J. (2007). Estimating the Estrada

index. Linear Algebra and its Applications, 427:70-76.

Dobrynin, A.A., Entringer, R. & Gutman, I. (2001). Wiener index

of trees: theory and applications. Acta Applicandae Mathematicae,


Estrada, E. (2000). Characterization of 3D molecular structure.

Chemical Physics Letters, 319:713-718.

Estrada, E. (2002). Characterization of the folding degree of proteins.

Bioinformatics, 18:697-704.

Estrada, E. (2004). Characterization of the amino acid contribution to

the folding degree of proteins. Proteins, 54:727-737.

Estrada, E. & Rodríguez-Velázquez, J.A. (2005). Spectral measures

of bipartivity in complex networks. Physical Review E, 72:046105.

Güngör, A.D. & Bozkurt, Ş.B. (2009). On the distance Estrada index of

graphs. Hacettepe Journal of Mathematics and Statistics, 38:277-283.

Gutman, I., Deng, H. & Radenković, S. (2011). The Estrada index: an

updated survey. In: D. Cvetković, I. Gutman (Eds.), Selected Topics on

Applications of Graph Spectra, Math. Inst., Beograd, 155-174.

Kroto, H.W., Heath, J.R., O’Brien, S.C., Curl, R.F. & Smalley, R.E.

(1985). C60: Buckminsterfullerene. Nature, 318:162-163.

Hilano, T. & Nomura, K. (1984). Distance degree regular graphs.

Journal of Combinatorial Theory, Series B, 37:96-100.

Ilić, A. & Stevanović, D. (2010). The Estrada index of chemical trees.

Journal of Mathematical Chemistry, 47:305-314.

Indulal, G. (2009). Sharp bounds on the distance spectral radius and

the distance energy of graphs. Linear Algebra and its Applications,


Indulal, G. & Gutman, I. (2008). On the distance spectra of some

graphs. Mathematical Communications, 13:123-131.

Jin, Y. & Zhang, X. (2014). Complete multipartite graphs are determined

by their distance spectra. Linear Algebra and its Applications, 448:285-291.

Nikolić, S., Trinajstić, N. & Mihalić, Z. (1995). The Wiener index:

development and applications. Croatica Chemica Acta, 68:105-129.

Randić, M. (1993). In search for graph invariants of chemical interest.

Journal of Molecular Structure, 300:551-571.

Shang, Y. (2015a). Distance Estrada index of random graphs. Linear

and Multilinear Algebra, 63:466-471.

Shang, Y. (2015b). The Estrada index of evolving graphs. Applied

Mathematics and Computation, 250:415-423.

Shang, Y. (2011). Perturbation results for the Estrada index in weighted

networks. Journal of Physics A: Mathematical and Theoretical,


Shang, Y. (2013). Lower bounds for the Estrada index using mixing

time and Laplacian spectrum. Rocky Mountain Journal of Mathematics,


Walikar, H.B., Shigehalli, V.S. & Ramane, H.S. (2004). Bounds on the

Wiener number of a graph. MATCH Communications in Mathematical

and in Computer Chemistry, 50:117-132.

Wiener, H. (1947). Structural determination of paraffin boiling points.

Journal of the American Chemical Society, 69:17-20.

Zhou, B. (2008). On Estrada index. MATCH Communications in

Mathematical and in Computer Chemistry, 60:485-492.

Zhou, B., Gutman, I. & Aleksić, T. (2008). A note on Laplacian energy

of graphs. MATCH Communications in Mathematical and in Computer Chemistry, 60:441-446.


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