Abstract
In a simple graph, Laplacian matrix and signless Laplacian matrix are derived from both adjacency matrix and degree matrix. Although, determinant of Laplacian matrix is always zero, yet we express it using only the adjacency matrix and square of its adjacency matrix. Likewise, we express the determinant of signless Laplacian matrix using only the diagonal elements provided the signless Laplacian matrix is equal to the square of its adjacency matrix.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bapat, R., Kirkland, S.J., Neumann, M.: On distance matrices and laplacians. Linear Algebra Appl. 401, 193–209 (2005)
Bapat, R.B.: On minors of the compound matrix of a laplacian. Linear Algebra Appl. 439(11), 3378–3386 (2013)
Biggs, N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1993)
Chung, F., Lu, L., Vu, V.: Spectra of random graphs with given expected degrees. P. Natl Acad. Sci. 100(11), 6313–6318 (2003)
Cvetkovic, D.: New theorems for signless laplacians eigenvalues. Bull. Acad. Serbe Sci. Arts Cl. Sci. Math. Natur. Sci. Math 137(33), 131–146 (2008)
Cvetkovic, D., Rowlinson, P., Simic, S.K.: Signless laplacians of finite graphs. Linear Algebra Appl. 423(1), 155–171 (2007)
De Abreu, N.M.M.: Old and new results on algebraic connectivity of graphs. Linear Algebra Appl. 423(1), 53–73 (2007)
Godsil, C.D., Royle, G.: Algebraic Graph Theory. Springer, New York (2001)
Goldberg, F., Kirkland, S.: On the sign patterns of the smallest signless laplacian eigenvector. Linear Algebra Appl. 443, 66–85 (2014)
Gutman, I., Lee, S., Chu, C., Luo, Y.: Chemical applications of the laplacian spectrum of molecular graphs: studies of the wiener number. Indian J. Chem. A. 33, 603–608 (1994)
Mesbahi, M., Egerstedt, M.: Graph Theoretic Methods in Multiagent Networks. Princeton University Press, Princeton (2010)
Mohar, B., Alavi, Y., Chartrand, G., Oellermann, O.: The laplacian spectrum of graphs. Graph Theory Comb. Appl. 2, 871–898 (1991)
Spielman, D.A.: Algorithms, graph theory, and linear equations in laplacian matrices. Proc. Int. Congr. Mathe. 4, 2698–2722 (2010)
Tam, B., Huang, P.: Nonnegative square roots of matrices. Linear Algebra Appl. 498, 404–440 (2016)
Teufl, E., Wagner, S.: Determinant identities for laplace matrices. Linear Algebra Appl. 432(1), 441–457 (2010)
Van Dam, E.R., Haemers, W.H.: Which graphs are determined by their spectrum? Linear Algebra Appl. 373, 241–272 (2003)
Von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)
West, D.B.: Introduction to Graph Theory. N.J. Prentice Hall, Upper Saddle River (2001)
Zelazo, D., Burger, M.: On the definiteness of the weighted laplacian and its connection to effective resistance. In: 53rd IEEE Conference on Decision and Control, pp. 2895–2900 (2014)
Zhang, X.: The laplacian eigenvalues of graphs: a survey. Linear Algebra Research Advances p. arXiv preprint (2011). arXiv:1111.2897
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Babarinsa, O., Kamarulhaili, H. (2017). On Determinant of Laplacian Matrix and Signless Laplacian Matrix of a Simple Graph. In: Arumugam, S., Bagga, J., Beineke, L., Panda, B. (eds) Theoretical Computer Science and Discrete Mathematics. ICTCSDM 2016. Lecture Notes in Computer Science(), vol 10398. Springer, Cham. https://doi.org/10.1007/978-3-319-64419-6_28
Download citation
DOI: https://doi.org/10.1007/978-3-319-64419-6_28
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-64418-9
Online ISBN: 978-3-319-64419-6
eBook Packages: Computer ScienceComputer Science (R0)