Nordhaus-Gaddum-type relations for the energy and Laplacian energy of graphs

Bulletin, Classe des Sciences Mathematiques et Naturelles, Sciences Mathematiques

Volumn 134, Issue 32, 2007, Pages 1-11

  Zhou, B ^a   Gutman, I ^b  

^a SOUTH CHINA NORMAL UNIVERSITY   (China)

^b UNIVERSITY OF KRAGUJEVAC   (Serbia)

Author keywords

Energy (of graph); Laplacian energy (of graph); Laplacian spectrum (of graph); Nordhaus Gaddum type relation; Spectrum (of graph)

Indexed keywords

EID: 51849113866     PISSN: 05617332     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (37)

1. N. Achuthan, N. R. Achuthan, L. Caccetta, On the Nordhaus-Gaddum class problem, Australas. J. Combin. 2 (1990) 5-27.
2. Y. Alavi, M. Behzard, Complementary graphs and edge chromatic numbers, SIAM J. Appl. Math. 20 (1971) 161-163.
3. R. Balakrishnan, The energy of a graph, Lin. Algebra Appl. 387 (2004) 287-295.
4. R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bull. Kerala Math. Assoc. 1 (2004) 129-132.
5. G. Chartrand, S. Schuster, On the independence numbers of complementary graphs, Trans. New York Acad. Sci. Ser. II 36 (1974) 247-251.
6. D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs - Theory and Application, Academic Press, New York, 1980; 2nd revised ed.: Barth, Heidelberg, 1995.
7. M. Fiedler, Algebraic connectivity of graphs, Czechoslovak Math. J. 25 (1975) 607-618.
8. W. Goddard, M. A. Henning, Nordhaus-Gaddum bounds for independent domination, Discr. Math. 268 (2003) 299-302.
9. W. Goddard, M. A. Henning, H. C. Swart, Some Nordhaus-Gaddum-type results, J. Graph Theory 16 (1992) 221-231.
10. R. Grone, R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discr. Math. 7 (1994) 221-229.
11. R. Grone, R. Merris, V. S. Sunder, The Laplacian spectrum of a graph, SIAM J. Matrix Anal. Appl. 11 (1990) 218-238.
12. I. Gutman, The energy of a graph: old and new results, in: A. Betten, A. Kohnert, R. Laue, A.Wassermann (Eds.), Algebraic Combinatorics and Applications, Springer-Verlag, Berlin, 2001, pp. 196-211.
13. I. Gutman, Topology and stability of conjugated hydrocarbons. The dependence of total π-electron energy on molecular topology, J. Serb. Chem. Soc. 70 (2005) 441-456.
14. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer-Verlag, Berlin, 1986.
15. I. Gutman, B. Zhou, Laplacian energy of a graph, Lin. Algebra Appl. 414 (2006) 29-37.
16. F. Harary, T. W. Haynes, Nordhaus-Gaddum inequalities for domination in graphs, Discr. Math. 155 (1996) 99-105.
17. Y. Hong, J. Shu, A sharp upper bound for the spectral radius of the Nordhas-Gaddum type, Discr. Math. 211 (2000) 229-232.
18. J. H. Koolen, V. Moulton, Maximal energy graphs, Adv. Appl. Math. 26 (2001) 47-52.
19. J. H. Koolen, V. Moulton, I. Gutman, Improving the McCelland inequality for total π-electron energy, Chem. Phys. Lett. 320 (2000) 213-216.
20. H. Liu, M. Lu, F. Tian, On the ordering of trees with the general Randić index of the Nordhaus-Gaddum type, MATCH Commun. Math. Comput. Chem. 55 (2006) 419-426.
21. R. Merris, Laplacian matrices of graphs: A survey, Lin. Algebra Appl. 197-198 (1994) 143-176.
22. B. Mohar, The Laplacian spectrum of graphs, in: Y. Alavi, G. Chartrand, O. R. Oellermann, A. J. Schwenk (Eds.), Graph Theory, Combinatorics, and Applications, Wiley, New York, 1991, pp. 871-898.
23. E. A. Nordhaus, J. W. Gaddum, On complementary graphs, Amer. Math. Monthly 63 (1956) 175-177.
24. E. Nosal, Eigenvalues of Graphs, M. Sc. thesis, Univ. Calgary, 1970.
25. J. Rada, Energy ordering of catacondensed hexagonal systems, Discr. Appl. Math. 145 (2005) 437-443.
26. J. Rada, A. Tineo, Upper and lower bounds for energy of bipartite graphs, J. Math. Anal. Appl. 289 (2004) 446-455.
27. H. S. Ramane, I. Gutman, H. B. Walikar, S. B. Halkarni, Another class of equienergetic graphs, Kragujevac J. Math. 26 (2004) 15-18.
28. H. S. Ramane, H. B. Walikar, S. B. Rao, B. D. Acharya, P. R. Hampiholi, S. R. Jog, I. Gutman, Equienergetic graphs, Kragujevac J. Math. 26 (2004) 5-13.
29. H. S. Ramane, H. B. Walikar, S. B. Rao, B. D. Acharya, P. R. Hampiholi, S. R. Jog, I. Gutman, Spectra and energies of iterated line graphs of regular graphs, Appl. Math. Lett. 18 (2005) 679-682.
30. I. Shparlinski, On the energy of some circulant graphs, Lin. Algebra Appl. 414 (2006) 378-382.
31. M. Stencel, Theorems of Nordhaus-Gaddum type for some numbers, Discuss. Math. 7 (1985) 107-112.
32. D. Stevanović, Energy and NEPS of graphs, Lin. Multilin. Algebra 53 (2005) 67-74.
33. D. Stevanović, I. Stanković, Remarks on hyperenergetic circulant graphs, Lin. Algebra Appl. 400 (2005) 345-348.
34. M. Stiebitz, On Hadwiger's number - a problem of the Nordhaus-Gaddum type, Discr. Math. 101 (1992) 307-317.
35. W. Yan, L. Ye, On the minimal energy of trees with a given diameter, Appl. Math. Lett. 18 (2005) 1046-1052.
36. L. Zhang, B. Wu, The Nordhaus-Gaddum-type inequalities for some chemical indices, MATCH Commun. Math. Comput. Chem. 54 (2005) 189-194.
37. B. Zhou, I. Gutman, On Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 57 (2007) 211-220.