Covering energy of posets and its bounds

Bhamre, Vandana P. ; Pawar, Madhukar M.

Mathematica Bohemica, Tome 148 (2023) no. 4, pp. 537-553

Abstract (VO)

The concept of covering energy of a poset is known and its McClelland type bounds are available in the literature. In this paper, we establish formulas for the covering energy of a crown with $2n$ elements and a fence with $n$ elements. A lower bound for the largest eigenvalue of a poset is established. Using this lower bound, we improve the McClelland type bounds for the covering energy for some special classes of posets.

Détail

DOI : 10.21136/MB.2022.0029-22

Classification : 05B20, 05C50, 06A07, 06A11, 06B05, 06B99
Keywords: covering energy of poset; eigenvalue; spectrum; upper bound; lower bound

Bibliographie

[1] Adiga, C., Bayad, A., Gutman, I., Srinivas, S. A.: The minimum covering energy of a graph. Kragujevac J. Sci. 34 (2012), 39-56.

[2] ndağ, Ş. B. Altı, Bozkurt, D.: Lower bounds for the energy of (bipartite) graphs. MATCH Commun. Math. Comput. Chem. 77 (2017), 9-14. | MR | JFM

[3] Collatz, L., Sinogowitz, U.: Spektren endlicher Grafen. Abh. Math. Semin. Univ. Hamb. 21 (1957), 63-77 German. | DOI | MR | JFM

[4] Cvetković, D. M., Doob, M., Sachs, H.: Spectra of Graphs: Theory and Application. Academic Press, New York (1980). | MR | JFM

[5] Das, K. C., Mojallal, S. A., Gutman, I.: Improving McClelland's lower bounds for energy. MATCH Commun. Math. Comput. Chem. 70 (2013), 663-668. | MR | JFM

[6] Davey, B. A., Priestley, H. A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990). | DOI | MR | JFM

[7] Grätzer, G.: General Lattice Theory. Pure and Applied Mathematics 75. Academic Press, New York (1978). | DOI | MR | JFM

[8] Gutman, I.: The energy of a graph. Ber. Math.-Stat. Sekt. Forschungszent. Graz 103 (1978), 22 pages. | MR | JFM

[9] Gutman, I., Furtula, B.: The total $\pi$-electron energy saga. Croat. Chem. Acta 90 (2017), 359-368. | DOI

[10] Gutman, I., Furtula, B.: Energies of Graphs: Survey, Census, Bibliography. Center for Scientific Research, Kragujevac (2019).

[11] Gutman, I., Ramane, H. S.: Research on graph energies in 2019. MATCH Commun. Math. Comput. Chem. 84 (2020), 277-292. | MR

[12] Hückel, E.: Quantentheoretische Beiträge zum Benzolproblem I. Die Electronenkonfiguration des Benzols und verwandter Verbindungen. Z. Phys. 70 (1931), 204-286 German. | DOI | JFM

[13] Indulal, G., Vijayakumar, A.: A note on energy of some graphs. MATCH Commun. Math. Comput. Chem. 59 (2008), 269-274. | MR | JFM

[14] Kelly, D., Rival, I.: Crowns, fences, and dismantlable lattices. Can. J. Math. 27 (1974), 1257-1271. | DOI | MR | JFM

[15] Li, X., Shi, Y., Gutman, I.: Graph Energy. Springer, Berlin (2012). | DOI | MR | JFM

[16] McClelland, B. J.: Properties of the latent roots of a matrix: The estimation of $\pi$-electron energies. J. Chem. Phys. 54 (1971), 640-643. | DOI

[17] Pawar, M. M., Bhamare, V. P.: On covering energy of posets. Math. Sci. Int. Research J. 4 (2015), 121-125.

[18] Pawar, M. M., Bhamre, V. P.: Covering energy of some classes of posets. J. Indian Math. Soc., New Ser. 87 (2020), 193-205. | DOI | MR | JFM

[19] Pawar, M. M., Bhangale, S. T.: Minimum independent dominating energy of graphs. Asian-Eur. J. Math. 14 (2021), Article 2150127, 14 pages. | DOI | MR | JFM

[20] Rival, I.: Lattices with doubly irreducible elements. Can. Math. Bull. 17 (1974), 91-95. | DOI | MR | JFM

[21] Thakare, N. K., Pawar, M. M., Waphare, B. N.: A structure theorem for dismantlable lattices and enumeration. Period. Math. Hung. 45 (2002), 147-160. | DOI | MR | JFM