A note on the bounds of Laplacian-energy-like-invariant

Document Type: Research Paper

Authors

1 payame noor university

2 Inviting lecturer of Iran university of science and technology

Abstract

The Laplacian-energy-like of a simple connected graph G is defined as
LEL:=LEL(G)=∑_(i=1)^n√(μ_i ),
Where μ_1 (G)≥μ_2 (G)≥⋯≥μ_n (G)=0 are the Laplacian eigenvalues of the graph G. Some upper and lower bounds for LEL are presented in this note. Moreover, throughout this work, some results related to lower bound of spectral radius of graph are obtained using the term of ΔG as the number of triangles in graph.

Keywords

Main Subjects


1. R. Bhatia, Matrix Analysis, Springer, New York, 1996.
2. B. Bollobás, Extremal Graph Theory, Academic Press, London, 1978.
3. M. Daianu, A. Mezher, N. Jahanshad, D. P. Hibar, T. M. Nir, C. R. Jack,
M. W. Weiner, M. A. Bernstein and P. M. Thompson, Spectral graph
theory and graph energy metrics show evidence for the Alzheimers disease
disconnection syndrome in APOE-4 risk gene carriers, Proc. IEEE Int.
Symp. Biomed Imaging 2015 (2015) 458−461.
4. K. C. Das, K. Xu and I. Gutman, Comparison between Kirchhoff index and
Laplacian-energy-like invariant of graphs, Linear Algebra Appl. 436 (9)
(2012) 3661−3671.

5. M. Dehmer, X. Li and Y. Shi, Connections between generalized graph
entropies and graph energy, Complexity 21 (1) (2015)35−41.
6. L. Di Paola, G. Mei, A. Di Venere and A. Giuliani, Exploring the stability
of dimers through protein structure topology, Curr. Protein Pept. Sci. 17
(1) (2016) 30−36.
7. M. Fiedler, Algebraic connectivity of graphs, Czech. Math. J. 23 (98)
(1973) 298−305.
8. D. C. Fisher, Lower bounds on the number of triangles in a graph, J. Graph
Theory 13 (4) (1989) 505−512.
9. I. Gutman, Estimating the Laplacian energy-like molecular structure
descriptor, Z. Naturforsch. 67 (2012) 403−406.
10. I. Gutman, The energy of a graph: old and new results, Algebraic
combinatorics and applications (Gweinstein, 1999), 196−211, Springer,
Berlin, 2001.
11. I. Gutman, X. Li and J. Zhang, Graph Energy, in: M. Dehmer, F. Emmert-
Streib (Eds.), Analysis of Complex Networks. From Biology to Linguistics,
Wiley-VCH, Weinheim, 2009, pp. 145−174.
12. I. Gutman, B. Zhou and B. Furtula, The Laplacian-energy like invariant is
an energy like invariant, MATCH Commun. Math. Comput. Chem. 64 (1)
(2010) 85−96.
13. Y. Hong, A bound on the spectral radius of graphs, Linear Algebra Appl.
108 (1988) 135−139.
14. A. Ilić, D. Krtinić and M. Ilić, On Laplacian like energy of trees, MATCH
Commun. Math. Comput. Chem. 64 (1) (2010) 111−122.
15. X. Li, Z. Qin, M. Wei, I. Gutman and M. Dehmer, Novel inequalities for
generalized graph entropies-graph energies and topological indices, Appl.
Math. Comput. 259 (2015) 470−479.
16. X. Li, Y. Shi and I. Gutman, Graph Energy, Springer, New York, 2012.
17. B. Liu, Y. Huang and Z. You, A survey on the Laplacian-energy-like
invariant, MATCH Commun. Math. Comput. Chem. 66 (3) (2011) 713−730.
18. J. Liu and B. Liu, A Laplacian-energy-like invariant of a graph, MATCH
Commun. Math. Comput. Chem. 59 (2) (2008) 355−372.
19. L. Lovász and M. Simonovits, On complete subgraphs of a graph II,
Studies Pure Math. (1983) 459−496.
20. R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl.
197−198 (1994) 143−176.

21. V. S. Nikiforov and N. G. Khadzhiivanov, Solution of the problem of P.
Erdos on the number of triangles in graphs with 􀝊 vertices and 􀝊􀬶/4 + 􀝈
edges, C. R. Acad. Bulg. Sci. 34 (1981) 969−970.
22. E. A. Nordhaus and B. M. Stewart, Triangles in an ordinary graph. Canad.
J. Math. 15 (1963) 33−41.
23. R. P. Stanley, A bound on the spectral radius of graphs with e edges,
Linear Algebra Appl. 87 (1987) 267−269.
24. D. Sun, C. Xu and Y. Zhang, A novel method of 2D graphical
representation for proteins and its application, MATCH Commun. Math.
Comput. Chem. 75 (2) (2016) 431−446.
25. S. W. Tan, On the Laplacian coefficients and Laplacian-like energy of
bicyclic graphs, Linear Multilinear Algebra 60 (9) (2012) 1071−1092.
26. P. Van Mieghem and R. van de Bovenkamp, Accuracy criterion for the
meaneld approximation in susceptible-infected-susceptible epidemics on
networks, Phys. Rev. E 91 (2015) 032812.
27. W. Wang and Y. Luo, On Laplacian-energy-like invariant of a graph,
Linear Algebra Appl. 437 (2) (2012) 713−721.
28. H. Wu, Y. Zhang, W. Chen and Z. Mu, Comparative analysis of protein
primary sequences with graph energy, Phys. A. 437 (2015) 249−262.
29. J. Zhang and J. Li, New results on the incidence energy of graphs, MATCH
Commun. Math. Comput. Chem. 68 (3) (2012) 777−803.
30. B. Zhou, More upper bounds for the incidence energy, MATCH Commun.
Math. Comput. Chem. 64 (1) (2010) 123−128.
31. B.-X. Zhu, The Laplacian-energy like of graphs, Appl. Math. Lett. 24 (9)
(2011) 1604−1607.