×

On the Estrada index of cactus graphs. (English) Zbl 1332.05089

Summary: The Estrada index of a graph \(G\), introduced by E. Estrada [“Characterization of 3D molecular structure”, Chem. Phys. Lett. 319, 713–718 (2000)], is defined as \(\mathit{EE}(G) = \sum_{i = 1}^n e^{\lambda_i}\), where \(\lambda_1, \lambda_2, \ldots, \lambda_n\) are the eigenvalues of a graph \(G\). A cactus graph is a connected graph in which any two simple cycles have at most one vertex in common. In this paper, we investigate cactus graphs in which every block is a triangle. The upper and lower bounds for Estrada index of these cactus graphs are obtained, and all the graphs attaining upper and lower bounds are characterized, respectively. The lower bound for Estrada index of these cactus graphs with given maximum degree is also obtained, and graph attaining lower bound is characterized.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C40 Connectivity
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bandad, 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] Cvetkoić, D.; Doob, M.; Sachs, H., Spectra of Graphs-Theory and Application (1980), Academic Press: Academic Press New York
[3] Das, K. C.; Lee, S.-G., On the Estrada index conjecture, Linear Algebra Appl., 431, 1351-1359 (2009) · Zbl 1175.05080
[4] de la Peña, J. A.; Gutman, I.; Rada, J., Estimating the Estrada index, Linear Algebra Appl., 427, 70-76 (2007) · Zbl 1184.05082
[5] Deng, H., A proof of a conjectures on the Estrada index, MATCH Commun. Math. Comput. Chem., 62, 599-606 (2009) · Zbl 1224.05295
[6] Deng, Q.; Chen, H., On extremal bipartite unicyclic graphs, Linear Algebra Appl., 444, 89-99 (2014) · Zbl 1292.05144
[7] Du, Z., An edge grafting theorem on the Estrada index of graphs and its applications, Discrete Appl. Math., 161, 134-139 (2013) · Zbl 1254.05100
[8] Du, Z.; Zhou, B., The Estrada index of trees, Linear Algebra Appl., 435, 2462-2467 (2011) · Zbl 1222.05022
[9] Du, Z.; Zhou, B.; Xing, R., On maximum Estrada indices of graphs with given parameters, Linear Algebra Appl., 436, 3767-3772 (2012) · Zbl 1241.05075
[10] Estrada, E., Characterization of 3D molecular structure, Chem. Phys. Lett., 319, 713-718 (2000)
[11] Estrada, E., Characterization of the amino acid contration to the folding degree of proteins, Proteins, 54, 727-737 (2004)
[12] Estrada, E., Characterization of the folding degree of proteins, Bioinformatics, 18, 697-704 (2002)
[13] Estrada, E., Topological structural classes of complex networks, Phys. Rev. E., 75, Article 016103 pp. (2007)
[14] Estrada, E.; Rodríguez-Velázquez, J. A., Subgraph centrality in complex networks, Phys. Rev. E., 71, 056103-1-9 (2005)
[15] Estrada, E.; Rodríguez-Velázquez, J. A., Subgraph centrality and clustering in complex hyper-networks, Physica A, 364, 581-594 (2006)
[16] Fath-Tabar, G. H.; Ashrafi, A. R., New upper bounds for Estrada index of bipartite graphs, Linear Algebra Appl., 435, 2607-2611 (2011) · Zbl 1222.05157
[17] Gutman, I., Lower bounds for Estrada index, Publ. Inst. Math., 83, 1-7 (2008) · Zbl 1199.05219
[18] 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
[19] Li, F.; Wei, L.; Cao, J., On the maximum Estrada index of 3-uniform linear hypertrees, Scientific World J., 8, 1-8 (2014)
[20] llic̀, A.; Stevanović, D., The Estrada index of chemical trees, J. Math. Chem., 47, 305-314 (2010) · Zbl 1309.92085
[21] Shang, Y., Perturbation results for the Estrada index in weighted networks, J. Phys. A, 44, 7, Article 075003 pp. (2011) · Zbl 1228.05264
[22] Shang, Y., Random lifts of graphs: Network robustness based on the Estrada index, Appl. Math. E Note., 12, 53-61 (2012) · Zbl 1253.05091
[23] Shang, Y., Local natural connectivity in complex networks, Chin. Phys. Lett., 28, 6, Article 068903 pp. (2011)
[24] Shang, Y., Biased edge failure in scale-free networks based on natural connectivity, Indian J. Phys., 86, 6, 485-488 (2012)
[25] Shang, Y., Lower bounds for the Estrada index using mixing time and Laplacian spectrum, Rocky Mountain J. Math., 43, 6, 2009-2016 (2013) · Zbl 1345.05042
[26] Shang, Y., Lower bounds for the Estrada index of graphs, Electron. J. Linear Algebra, 23, 664-668 (2012) · Zbl 1250.15015
[27] Shang, Y., Estrada index of general weighted graphs, Bull. Aust. Math. Soc., 88, 1, 106-112 (2013) · Zbl 1271.05057
[28] Wang, W.; Xu, W., Graphs with the maximal Estrada indices, Linea Algebra Appl., 446, 314-328 (2014) · Zbl 1292.05181
[29] Zhao, H.; Jia, Y., On the Estrada index of bipartite graph, MATCH Commun. Math. Comput. Chem., 61, 495-501 (2009) · Zbl 1193.92107
[30] Zhou, B., On Estrada index, MATCH. Commun. Match. Comput. Chem., 60, 485-492 (2008) · Zbl 1199.05254
[31] Zhou, B.; Trinajstić, N., Estrada index of bipartite graphs, Int. J. Chem. Model., 1, 387-394 (2008)
[32] Zhu, Z.; Tan, L.; Qiu, Z., Tricyclic graph with maximal Estrada index, Discrete Appl. Math., 16, 364-372 (2014) · Zbl 1300.05166
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.