Albertson energy and Albertson Estrada index of graphs
الموضوعات :
1 - Department of Mathematics, Shahrood University of Technology, Shahrood, Iran
الکلمات المفتاحية: energy, Eigenvalue of graph, Albertson Estrada index, Albertson energy,
ملخص المقالة :
Let $G$ be a graph of order $n$ with vertices labeled as $v_1, v_2,\dots , v_n$. Let $d_i$ be the degree of the vertex $v_i$ for $i = 1, 2, \cdots , n$. The Albertson matrix of $G$ is the square matrix of order $n$ whose $(i, j)$-entry is equal to $|d_i - d_j|$ if $v_i $ is adjacent to $v_j$ and zero, otherwise. The main purposes of this paper is to introduce the Albertson energy and Albertson-Estrada index of a graph, both base on the eigenvalues of the Albertson matrix. Moreover, we establish upper and lower bounds for these new graph invariants and relations between them.
[1] H. Abdo, D. Dimitrov, The total irregularity of graphs under graph operations. Miskolc Math. Notes. 15 (2014), 3-17.
[2] M. O. Albertson, The irregularity of a graph, Ars Combin. 46 (1997), 219-225.
[3] R. Balakrishnan, The energy of a graph, Linear. Algebra. Appl. 387 (2004), 287-295.
[4] Z. Cvetkovskic, Inequalities, Theorems, Techniques and Selected problems, Springer-Verlag, Berlin, 2012.
[5] D. Cvetkovic, P. Rowlinson, S. Simic, An Introduction to the Theory of Graph Spectra, Cambridge University Press, Cambridge, 2010.
[6] K. C. Das, Maximizing the sum of the squares of the degrees of a graph, Discr. Math. 285 (2004), 57-66.
[7] K. C. Das, I. Gutman, Bounds for the energy of graphs, Hacettepe J. Math. Stat., in press.
[8] K. C. Das, I. Gutman, B. Zhou, New upper bounds on Zagreb indices, J. Math. Chem. 46 (2009), 514-521.
[9] J. A. De la Pena, I. Gutman, J. Rada, Estimating the Estrada Index, Linear. Algebra. Appl. 427 (2007), 70-76.
[10] H. Deng, S. Radenkovic, I. Gutman, The Estrada Index, Applications of Graph Spectra, Belgrade, 2009.
[11] E. Estrada, Characterization of the amino acid contribution to the folding degree of proteins, Proteins Struct. Funct. Bioinf. 54 (2004), 727-737.
[12] E. Estrada, Characterization of the folding degree of proteins, Bioinf. 18 (2002), 697-704.
[13] G. H. Fath-Tabar, Old and new Zagreb indices of graphs, Match. Commun. Math. Comput. Chem. 65 (2011), 79-84.
[14] W. Gao, M. R. Farahani, L. Shi, Forgotten topological index of some drug structures, Acta. Med. Mediter. (2016), 2016:579.
[15] W. Gao, W. Wang, M. R. Farahani, Topological indices study of molecular structure in anticancer drugs, J. Chem. (2016), 2016:3216327.
[16] I. Gutman, Bounds for total electron energy, Chem. Phys. Lett. 24 (1974), 283-285.
[17] I. Gutman, Degree-based topological indices, Croat. Chem. Acta. (2013), 2013:2294.
[18] I. Gutman, McClelland-type lower bound for total electron energy, J. Chem. Soc. Faraday Trans. 86 (1990), 3373-3375.
[19] I. Gutman, The energy of a graph, Ber. Math. Statist. Sekt. Forschungsz. Graz. 103 (1978), 1-22.
[20] I. Gutman, K. C. Das, The first Zagreb index 30 years after, Match. Commun. Math. Comput. Chem. 50 (2004), 83-92.
[21] I. Gutman, P. Hansen, H. Melot, Variable neighborhood search for extremal graphs 10. Comparision of irregularity indices for chemical trees, J. Chem. Inf. Model. (2005), 2005: 0342775.
[22] I. Gutman, N. Trinajstic, Graph theory and molecular orbitals. Total electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535-538.
[23] P. Hansen, H. Melot, Variable neighborhood search for extremal graphs 9. Bounding the irregularity of a graph, Graphs and Discovery, Am. Math. Soc. Providence, 2005.
[24] B. Horoldagva, L. Buyantogtokh, S. Dorjsembe, I. Gutman, Maximum size of maximally irregular graphs, Match. Commun. Math. Comput. Chem. 76 (2016), 81-98.
[25] Y. Huang. B. Liu, M. Zhang, On Comparing the variable Zagreb indices, Match. Commun. Math. Comput. Chem. 63 (2010), 453-460.
[26] B. Liu, Z. You, A survey on comparing Zagreb indices, Match. Commun. Math. Comput. Chem. 65 (2011), 581-593.
[27] N. Jafari Rad, A. Jahanbani, R. Hasni, Pentacyclic graphs with maximal Estrada index, Ars Combin. 133 (2017), 133-145.
[28] N. Jafari Rad, A. Jahanbani, I. Gutman, Zagreb energy and Zagreb estrada index of graphs, Match. Commun. Math. Comput. Chem. 79 (2018), 371-386.
[29] N. Jafari Rad, A. Jahanbani, D. A. Mojdeh, Tetracyclic graphs with maximal Estrada index, Discrete Mathematics. Algorithms and Applications. (2017), 2017: 1750041.
[30] A. Jahanbani, Lower bounds for the energy of graphs, AKCE Inter. J. Graphs. Combin. 15 (2018), 88-96.
[31] A. Jahanbani, New Bounds for the harmonic energy and harmonic estrada index of graphs, Computer Science. 26 (2018), 270-300.
[32] A. Jahanbani, Some new lower bounds for energy of graphs, Appl. Math. Comput. 296 (2017), 233-238.
[33] A. Jahanbani, Some new lower bounds for energy of graphs, Match. Commun. Math. Comput. Chem. 79 (2018), 275-286.
[34] A. Jahanbani, H. H. Raz, On the harmonic energy and Estrada index of graphs. MATI 1 (2019), 1-20. http://dergipark.gov.tr/mati/issue/38227/425047.
[35] H. Kamarulhaili N. Alawiah, N. J. Rad, A. Jahanbani, New upper bounds on the energy of a graph, Match. Commun. Math. Comput. Chem. 79 (2018), 287-301.
[36] K. Ruedenberg, Quantum mechanics of mobile electrons in conjugated bond systems. III. Topological matrix as generatrix of bond orders, J. Chem. Phys. 4 (1961), 1884-1891.
[37] K. Ruedenberg, Theorem on the mobile bond orders of alternant conjugated systems, J. Chem. Phys. 29 (1958), 1232-1233.
[38] Z. Yan, H. Liu, H. Liu, Sharp bounds for the second Zagreb index of unicyclic graphs, J. Math. Chem. 42 (2007), 565-574.
[39] B. Zhou, Remarks on Zagreb indices, Match. Commun. Math. Comput. Chem. 57 (2007) 597-616.
[40] B. Zhou, Upper bounds for the Zagreb indices and the spectral radius of seriesparallel graphs, Int. J. Quantum Chem. 107 (2007), 875-878.
[41] B. Zhou, Z. Du, Some lower bounds for Estrada index, Iran. J. Math. Chem. 1 (2010), 67-72.
[42] B. Zhou, I. Gutman, Relations between Wiener, hyper-Wiener and Zagreb indices, Chem. Phys. Lett. 394 (2004), 93-95.
