×

Graphs with the maximal Estrada indices. (English) Zbl 1292.05181

Summary: Two new transformations are proposed to compare the Estrada indices between two graphs. Let \({\Psi}_{n,m}\) be the set of the \((n,m)\)-graphs, where \(n\) and \(m\) are the numbers of vertices and edges, respectively. The graphs with the maximal Estrada indices in \({\Psi}_{n,m}\) are deduced by the new method for three cases, namely unicyclic and bipartite unicyclic graphs \((m=n)\), bicyclic graphs \((m=n+1)\), and the \((n,m)\)-graphs without even cycles \((n+1 \leqslant m\leqslant 3(n-1)/2)\).

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C35 Extremal problems in graph theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Cvetkovic, D. M.; Doob, M.; Sachs, H., Spectra of graphs theory and application, (1980), Academic Press New York · Zbl 0458.05042
[2] Estrada, E., Characterization of 3d molecular structure, Chem. Phys. Lett., 319, 713-718, (2000)
[3] Estrada, E.; Rodriguez-Velazquez, J. A., Subgraph centrality in complex networks, Phys. Rev. E, 71, 056103, (2005)
[4] Estrada, E., Characterization of the folding degree of proteins, Bioinform., 18, 697-704, (2002)
[5] Estrada, E., Characterization of the amino acid contribution to the folding degree of proteins, Proteins: Struct. Funct. Bioinform., 54, 727-737, (2004)
[6] Estrada, E., Topological structural classes of complex networks, Phys. Rev. E, 75, 016103, (2007)
[7] Gutman, I., Lower bounds for estrada index, Publ. Inst. Math., 83, 1-7, (2008) · Zbl 1199.05219
[8] de la Pena, J. A.; Gutman, I.; Rada, J., Estimating the estrada index, Linear Algebra Appl., 427, 70-76, (2007) · Zbl 1184.05082
[9] Zhou, B., On estrada index, MATCH Commun. Math. Comput. Chem., 60, 485-492, (2008) · Zbl 1199.05254
[10] 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
[11] 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
[12] Zhao, H.; Jia, Y., On the estrada index of bipartite graph, MATCH Commun. Math. Comput. Chem., 61, 495-501, (2009) · Zbl 1193.92107
[13] Gutman, I.; Radenković, S.; Furtula, B.; Mansour, T.; Schork, M., Relating estrada index with spectral radius, J. Serbian Chem. Soc., 72, 1321-1327, (2007)
[14] Wang, L.; Fan, Y. Z.; Wang, Y., Maximum estrada index of bicyclic graphs, (17 Apr. 2012)
[15] Zhu, Z.; Tan, L.; Qiu, Z., Tricyclic graph with maximal estrada index, Discrete Appl. Math., 162, 364-372, (2014) · Zbl 1300.05166
[16] 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
[17] 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
[18] Chen, Z.; Fan, Y.; Du, W., Estrada index of random graphs, MATCH Commun. Math. Comput. Chem., 68, 825-834, (2012) · Zbl 1289.05273
[19] Du, Z.; Liu, Z., On the estrada and Laplacian estrada indices of graphs, Linear Algebra Appl., 435, 2065-2076, (2011) · Zbl 1221.05211
[20] Li, J.; Zhang, J., On the Laplacian estrada index of unicyclic graph, MATCH Commun. Math. Comput. Chem., 68, 835-842, (2012) · Zbl 1289.05299
[21] Khosravanirad, A., A lower bound for Laplacian estrada index of a graph, MATCH Commun. Math. Comput. Chem., 70, 175-180, (2013) · Zbl 1299.05222
[22] Deng, H., A proof of a conjecture on the estrada index, MATCH Commun. Math. Comput. Chem., 62, 599-606, (2009) · Zbl 1224.05295
[23] Li, J.; Li, X.; Wang, L., The minimal estrada index of trees with two maximum degree vertices, MATCH Commun. Math. Comput. Chem., 64, 799-810, (2010) · Zbl 1265.05388
[24] Zhang, J.; Zhou, B.; Li, J., On estrada index of trees, Linear Algebra Appl., 434, 215-223, (2011) · Zbl 1231.05176
[25] Du, Z.; Zhou, B., The estrada index of trees, Linear Algebra Appl., 435, 2462-2467, (2011) · Zbl 1222.05022
[26] Du, Z.; Zhou, B., The estrada index of unicyclic graphs, Linear Algebra Appl., 436, 3149-3159, (2012) · Zbl 1241.05073
[27] Wang, W. H., The unicyclic graph with the maximal estrada indices, MATCH Commun. Math. Comput. Chem., 68, 939-955, (2012) · Zbl 1289.05320
[28] Du, Z.; Zhou, B., On the estrada index of graphs with given number of cut edges, Electron. J. Linear Algebra, 22, 586-592, (2011) · Zbl 1227.05179
[29] Ilic, A.; Stevanovic, D., The estrada index of chemical trees, J. Math. Chem., 47, 305-314, (2010) · Zbl 1309.92085
[30] Anuradha, A.; Balakrishnan, R.; So, W., Skew spectra of graphs without even cycles, Linear Algebra Appl., 444, 67-80, (2014) · Zbl 1292.05165
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.