×

On the Estrada and Laplacian Estrada indices of graphs. (English) Zbl 1221.05211

Summary: The Estrada index of a graph \(G\) is defined as \(EE(G)=\sum ^n_{i=1} e^{\lambda_i}\), where \(\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}\) are the eigenvalues of \(G\). The Laplacian Estrada index of a graph \(G\) is defined as \(LEE(G)=\sum ^n_{i=1} e^{\mu_i}\), where \(\mu _{1},\mu _{2},\ldots ,\mu _{n}\) are the Laplacian eigenvalues of \(G\). An edge grafting operation on a graph moves a pendent edge between two pendent paths. We study the change of Estrada index of graph under edge grafting operation between two pendent paths at two adjacent vertices. As the application, we give the result on the change of Laplacian Estrada index of bipartite graph under edge grafting operation between two pendent paths at the same vertex.
We also determine the unique tree with minimum Laplacian Estrada index among the set of trees with given maximum degree, and the unique trees with maximum Laplacian Estrada indices among the set of trees with given diameter, number of pendent vertices, matching number, independence number and domination number, respectively.

MSC:

05C35 Extremal problems in graph theory
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C05 Trees
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bamdad, H.; Ashraf, F.; Gutman, I., Lower bounds for Estrada index and Laplacian Estrada index, Appl. Math. Lett., 23, 739-742 (2010) · Zbl 1203.05090
[2] Bondy, J. A.; Murty, U. S.R., Graph Theory with Applications (1976), American Elsevier: American Elsevier New York · Zbl 1134.05001
[3] Cvetković, D.; Doob, M.; Sachs, H., Spectra of Graphs - Theory and Application (1980), Academic Press: Academic Press New York · Zbl 0458.05042
[4] Das, K. C.; Lee, S.-G., On the Estrada index conjecture, Linear Algebra Appl., 431, 1351-1359 (2009) · Zbl 1175.05080
[5] Deng, H., A proof of a conjecture on the Estrada index, Match, 62, 599-606 (2009) · Zbl 1224.05295
[6] Deng, H., A note on the Estrada index of trees, Match, 62, 607-610 (2009) · Zbl 1224.05094
[7] Deng, H.; Zhang, J., A note on the Laplacian Estrada index of trees, Match, 63, 777-782 (2010) · Zbl 1265.05362
[8] Z. Du, More on the Laplacian Estrada indices of trees, in press.; Z. Du, More on the Laplacian Estrada indices of trees, in press. · Zbl 1289.05280
[9] Z. Du, B. Zhou, The Estrada index of trees, Linear Algebra Appl., in press, doi:10.1016/j.laa.2011.01.041; Z. Du, B. Zhou, The Estrada index of trees, Linear Algebra Appl., in press, doi:10.1016/j.laa.2011.01.041 · Zbl 1222.05022
[10] Z. Du, B. Zhou, On the Estrada index of graphs with given number of cut edges, in press.; Z. Du, B. Zhou, On the Estrada index of graphs with given number of cut edges, in press. · Zbl 1227.05179
[11] Estrada, E., Characterization of 3D molecular structure, Chem. Phys. Lett., 319, 713-718 (2000)
[12] Estrada, E., Characterization of the folding degree of proteins, Bioinformatics, 18, 697-704 (2002)
[13] Estrada, E., Characterization of the amino acid contribution to the folding degree of proteins, Proteins, 54, 727-737 (2004)
[14] Estrada, E.; Rodríguez-Valázquez, J. A., Subgraph centrality in complex net-works, Phys. Rev. E, 71, 056103-1-056103-9 (2005)
[15] Estrada, E.; Rodríguez-Valázquez, J. A., Spectral measures of bipartivity in complex networks, Phys. Rev. E, 72, 046105-1-046105-6 (2005)
[16] Estrada, E.; Rodríguez-Valázquez, J. A.; Randić, M., Atomic branching in molecules, Int. J. Quantum Chem., 106, 823-832 (2006)
[17] de la Peña, J. A.; Gutman, I.; Rada, J., Estimating the Estrada index, Linear Algebra Appl., 427, 70-76 (2007) · Zbl 1184.05082
[18] Fath-Tabar, G. H.; Ashrafi, A. R.; Gutman, I., Note on Estrada and L-Estrada indices of graphs, Bull. Cl. Sci. Math. Nat. Sci. Math., 139, 1-16 (2009) · Zbl 1274.05292
[19] G.H. Fath-Tabar, A.R. Ashrafi, New upper bounds for Estrada index of bipartite graphs, Linear Algebra Appl., in press, doi:10.1016/j.laa.2011.01.034; G.H. Fath-Tabar, A.R. Ashrafi, New upper bounds for Estrada index of bipartite graphs, Linear Algebra Appl., in press, doi:10.1016/j.laa.2011.01.034 · Zbl 1222.05157
[20] Guo, J., The effect on the Laplacian spectral radius of a graph by adding or grafting edges, Linear Algebra Appl., 413, 59-71 (2006) · Zbl 1082.05059
[21] Gutman, I.; Estrada, E.; Rodríguez-Velázquez, J. A., On a graph-spectrum-based structure descriptor, Croat. Chem. Acta, 80, 151-154 (2007)
[22] Gutman, I.; Furtula, B.; Marković, V.; Glišić, B., Alkanes with greatest Estrada index, Z. Naturforsch. A, 62, 495-498 (2007)
[23] Gutman, I.; Deng, H.; Radenković, S., The Estrada index: an updated survey, (Cvetković, D.; Gutman, I., Selected Topics on Applications of Graph Spectra (2011), Math. Inst.: Math. Inst. Beograd), 155-174 · Zbl 1289.05288
[24] Ilić, A.; Stevanović, D., The Estrada index of chemical trees, J. Math. Chem., 47, 305-314 (2010) · Zbl 1309.92085
[25] Ilić, A.; Zhou, B., Laplacian Estrada index of trees, Match, 63, 769-776 (2010) · Zbl 1265.05379
[26] Li, J.; Li, X.; Wang, L., The minimal Estrada index of trees with two maximum degree vertices, Match, 64, 799-810 (2010) · Zbl 1265.05388
[27] Li, J., A note on the maximal Estrada index of trees with a given bipartition, Match, 66, 765-768 (2011) · Zbl 1265.05386
[28] Li, J.; Shiu, W.-C.; Chang, A., On Laplacian Estrada index of a graph, Appl. Anal. Discrete Math., 3, 147-156 (2009) · Zbl 1274.05296
[29] Li, J.; Shiu, W.-C.; Chan, W.-H., Note on the Laplacian Estrada index of a graph, Match, 66, 777-784 (2011) · Zbl 1265.05385
[30] Li, Q.; Feng, K., On the largest eigenvalue of a graph, Acta Math. Appl. Sinica, 2, 167-175 (1979)
[31] Merris, R., Laplacian matrices of graphs: a survey, Linear Algebra Appl., 197-198, 143-176 (1994) · Zbl 0802.05053
[32] Shan, H.; Shao, J.; Gong, F.; Liu, Y., An edge grafting theorem on the energy of unicyclic and bipartite graphs, Linear Algebra Appl., 433, 547-556 (2010) · Zbl 1216.05078
[33] Shan, H.; Shao, J., Graph energy change due to edge grafting operations and its applications, Match, 64, 25-40 (2010) · Zbl 1265.05412
[34] Zhang, J.; Zhou, B.; Li, J., On Estrada index of trees, Linear Algebra Appl., 434, 215-223 (2011) · Zbl 1231.05176
[35] Zhao, H.; Jia, Y., On the Estrada index of bipartite graphs, Match, 61, 495-501 (2009) · Zbl 1193.92107
[36] Zhou, B., On Estrada index, Match, 60, 485-492 (2008) · Zbl 1199.05254
[37] Zhou, B.; Trinajstić, N., Estrada index of bipartite graphs, Int. J. Chem. Model., 1, 3/4, 387-394 (2008)
[38] Zhou, B.; Gutman, I., More on the Laplacian Estrada index, Appl. Anal. Discrete Math., 3, 371-378 (2009) · Zbl 1199.05255
[39] Zhou, B., On sum of powers of Laplacian eigenvalues and Laplacian Estrada index of graphs, Match, 62, 611-619 (2009) · Zbl 1274.05084
[40] Zhu, B., On the Laplacian Estrada index of graphs, Match, 66, 769-776 (2011) · Zbl 1265.05439
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.