Czechoslovak Mathematical Journal, Vol. 71, No. 2, pp. 309-319, 2021


Remarks on the bounds of graph energy in terms of vertex cover number or matching number

Xiaodan Chen, Xiaoqian Liu

Received April 23, 2019.   Published online March 5, 2021.

Abstract:  We give a novel upper bound on graph energy in terms of the vertex cover number, and present a complete characterization of the graphs whose energy equals twice their matching number.
Keywords:  graph energy; vertex cover number; matching number; bound
Classification MSC:  05C50


References:
[1] S. Akbari, E. Ghorbani, S. Zare: Some relations between rank, chromatic number and energy of graphs. Discrete Math. 309 (2009), 601-605. DOI 10.1016/j.disc.2008.09.012 | MR 2499013 | Zbl 1194.05075
[2] E. Andrade, M. Robbiano, B. San Martín: A lower bound for the energy of symmetric matrices and graphs. Linear Algebra Appl. 513 (2017), 264-275. DOI 10.1016/j.laa.2016.10.022 | MR 3573802 | Zbl 1350.05090
[3] Ş. B. Bozkurt Altindağ, D. Bozkurt: Lower bounds for the energy of (bipartite) graphs. MATCH Commun. Math. Comput. Chem. 77 (2017), 9-14. MR 3645362
[4] G. Caporossi, D. Cvetković, I. Gutman, P. Hansen: Variable neighborhood search for extremal graphs 2. Finding graphs with extremal energy. J. Chem. Inf. Comput. Sci. 39 (1999), 984-996. DOI 10.1021/ci9801419
[5] B. Cheng, B. Liu: On the nullity of graphs. Electron. J. Linear Algebra 16 (2007), 60-67. DOI 10.13001/1081-3810.1182 | MR 2285832 | Zbl 1142.05336
[6] C. A. Coulson: On the calculation of the energy in unsaturated hydrocarbon molecules. Proc. Camb. Philos. Soc. 36 (1940), 201-203. DOI 10.1017/S0305004100017175
[7] D. Cvetković, P. Rowlinson, S. Simić: An Introduction to the Theory of Graph Spectra. London Mathematical Society Student Texts 75. Cambridge University Press, Cambridge (2010). DOI 10.1017/CBO9780511801518 | MR 2571608 | Zbl 1211.05002
[8] K. C. Das, S. A. Mojallal, I. Gutman: Improving McClelland's lower bound for energy. MATCH Commun. Math. Comput. Chem. 70 (2013), 663-668. MR 3155011 | Zbl 1299.05213
[9] J. Day, W. So: Graph energy change due to edge deletion. Linear Algebra Appl. 428 (2008), 2070-2078. DOI 10.1016/j.laa.2007.11.009 | MR 2401641 | Zbl 1136.05037
[10] R. A. Horn, C. R. Johnson: Matrix Analysis. Cambridge University Press, Cambridge (1985). DOI 10.1017/CBO9780511810817 | MR 0832183 | Zbl 0576.15001
[11] A. Jahanbani: Some new lower bounds for energy of graphs. Appl. Math. Comput. 296 (2017), 233-238. DOI 10.1016/j.amc.2016.10.019 | MR 3572791 | Zbl 1411.05165
[12] J. H. Koolen, V. Moulton: Maximal energy graphs. Adv. Appl. Math. 26 (2001), 47-52. DOI 10.1006/aama.2000.0705 | MR 1806691 | Zbl 0976.05040
[13] X. Li, Y. Shi, I. Gutman: Graph Energy. Springer, New York (2012). DOI 10.1007/978-1-4614-4220-2 | MR 2953171 | Zbl 1262.05100
[14] B. J. McClelland: Properties of the latent roots of a matrix: The estimation of $\oldpi$-electron energies. J. Chem. Phys. 54 (1971), 640-643. DOI 10.1063/1.1674889
[15] I. Milovanović, E. Milovanović, I. Gutman: Upper bounds for some graph energies. Appl. Math. Comput. 289 (2016), 435-443. DOI 10.1016/j.amc.2016.05.045 | MR 3515866 | Zbl 1410.05138
[16] J. Rada, A. Tineo: Upper and lower bounds for the energy of bipartite graphs. J. Math. Anal. Appl. 289 (2004), 446-455. DOI 10.1016/j.jmaa.2003.08.027 | MR 2026917 | Zbl 1034.05034
[17] L. Wang, X. Ma: Bounds of graph energy in terms of vertex cover number. Linear Algebra Appl. 517 (2017), 207-216. DOI 10.1016/j.laa.2016.12.015 | MR 3592020 | Zbl 1353.05082
[18] D. Wong, X. Wang, R. Chu: Lower bounds of graph energy in terms of matching number. Linear Algebra Appl. 549 (2018), 276-286. DOI 10.1016/j.laa.2018.03.040 | MR 3784349 | Zbl 1390.05139
[19] A. Yu, M. Lu, F. Tian: New upper bounds for the energy of graphs. MATCH Commun. Math. Comput. Chem. 53 (2005), 441-448. MR 2134203 | Zbl 1081.05067
[20] B. Zhou: Energy of a graph. MATCH Commun. Math. Comput. Chem. 51 (2004), 111-118. MR 2063930 | Zbl 1106.05068

Affiliations:   Xiaodan Chen (corresponding author), College of Mathematics and Information Science, Guangxi University, No. 100, Daxue Road, Nanning, Guangxi, P. R. China, and Guangxi Center for Mathematical Research, Guangxi University, No. 100, Daxue Road, Nanning, Guangxi, P. R. China, e-mail: x.d.chen@live.cn; Xiaoqian Liu, College of Mathematics and Information Science, Guangxi University, No. 100, Daxue Road, Nanning, Guangxi, P. R. China, e-mail: xqliu95@163.com


 
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