Abstract
In analogy to the absolute algebraic connectivity of Fiedler, we study the problem of minimizing the maximum eigenvalue of the Laplacian of a graph by redistributing the edge weights. Via semidefinite duality this leads to a graph realization problem in which nodes should be placed as close as possible to the origin while adjacent nodes must keep a distance of at least one. We prove three main results for a slightly generalized form of this embedding problem. First, given a set of vertices partitioning the graph into several or just one part, the barycenter of each part is embedded on the same side of the affine hull of the set as the origin. Second, there is an optimal realization of dimension at most the tree-width of the graph plus one and this bound is best possible in general. Finally, bipartite graphs possess a one dimensional optimal embedding.
Similar content being viewed by others
References
Belk M., Connelly R.: Realizability of graphs. Discrete Comput. Geom. 37(2), 125–137 (2007)
Biyikoglu, T., Leydold, J., Stadler, P.F. (eds): Laplacian Eigenvectors of Graphs, volume 1915 of Lecture Notes in Mathematics. Springer, Berlin (2007)
Chung, F.R.: Spectral graph theory. volume 92 of Regional Conference Series in Mathematics. American Mathematical Society (AMS), Providence, RI (1997)
Connelly R.: Tensegrity structures: Why are they stable?. In: Thorpe, M.F., Duxbury, P.M. (eds) Rigidity Theory and Applications, pp. 47–54. Kluwer/Plenum, New York (1999)
Connelly R.: Generic global rigidity. Discrete Comput. Geom. 33(4), 549–563 (2005)
Cvetković D., Doob M., Sachs H.: Spectra of Graphs. Theory and Application. 3rd edn. J.A. Barth, Leipzig (1995)
Diestel R.: Graph Theory. 3rd edn. Springer, Berlin (2006)
Feynman, R., Leighton, R., Sands, M.: Vorlesungen über Physik. Band I: Mechanik, Strahlung, Wärme. Oldenbourg Wissensch.Vlg; Auflage: 4., durchges. A (2001)
Fiedler M.: Laplacian of graphs and algebraic connectivity. Comb. Graph Theory 25, 57–70 (1989)
Fiedler M.: Absolute algebraic connectivity of trees. Linear Multilinear Algebra 26, 85–106 (1990)
Ghosh A., Boyd S., Saberi A.: Minimizing effective resistance of a graph. SIAM Rev. 50(1), 37–66 (2008)
Gineityte V., Gutman I., Lepovic M., Miroslav P.: The high-energy band in the photoelectron spectrum of alkanes and its dependence on molecular structure. J. Serbian Chem. Soc. 64(11), 673–680 (1999)
Göring F., Helmberg C., Wappler M.: Embedded in the shadow of the separator. SIAM J. Optim. 19(1), 472–501 (2008)
Göring, F., Helmberg, C., Wappler, M.: The rotational dimension of a graph. Preprint 2008–16, Fakultät für Mathematik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany, Oct 2008. (To appear) in J. Graph Theory
Gutman I.: Hyper-Wiener index and Laplacian spectrum. J. Serbian Chem. Soc. 68(12), 949–952 (2003)
Gutman I., Vidovic D., Stevanovic D.: Chemical applications of the Laplacian spectrum. VI. On the largest Laplacian eigenvalue of alkanes. J. Serbian Chem. Soc. 67(6), 407–413 (2002)
Hendrickson B.: The molecule problem: exploiting structure in global optimization. SIAM J. Optim. 5(4), 835–857 (1995)
Klenke A.: Probability theory. A comprehensive course. Universitext. Springer, London (2008)
Mohar, B.: The Laplacian spectrum of graphs. In: Graph Theory, Combinatorics, and Applications, pp. 871–898. Wiley, New York (1991)
Mohar B.: Graph Laplacians. In: Beineke, L.W., Wilson, R.J., Cameron, P.J. (eds) Topics in algebraic graph theory volume 102 of Encyclopedia of Mathematics and Its Applications, pp. 113–136. Cambridge University Press, Cambridge (2004)
Mohar B., Poljak S.: Eigenvalues and the max-cut problem. Czech. Math. J. 40(115), 343–352 (1990)
Nikiforov V.: Bounds on graph eigenvalues. I. Linear Algebra Appl. 420(2–3), 667–671 (2007)
Pendavingh, R.A.: Spectral and geometrical graph characterizations. Ph.D. thesis, Universiteit van Amsterdam, Amsterdam, Oct (1998)
Shi L.: Bounds on the (Laplacian) spectral radius of graphs. Linear Algebra Appl. 422(2–3), 755–770 (2007)
Solé P.: Expanding and forwarding. Discrete Appl. Math. 58(1), 67–78 (1995)
Sun J., Boyd S., Xiao L., Diaconis P.: The fastest mixing Markov process on a graph and a connection to a maximum variance unfolding problem. SIAM Rev. 48(4), 681–699 (2006)
van der Holst H., Lovász L., Schrijver A. et al.: The Colin de Verdière graph parameter. In: Lovász, L. (eds) Graph theory and combinatorial biology volume 7 of Bolyai Soc. Math. Stud., pp. 29–85. János Bolyai Mathematical Society, Budapest (1999)
von Luxburg U., Belkin M., Bousquet O.: Consistency of spectral clustering. Ann. Stat. 36(2), 555–586 (2008)
Wang T.-F.: Several sharp upper bounds for the largest Laplacian eigenvalue of a graph. Sci. China, Ser. A 50(12), 1755–1764 (2007)
Weinberger, K.Q., Saul, L.K.: Unsupervised learning of image manifolds by semidefinite programming. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR-04), Washington, DC, vol. 2, pp. 988–995 (2004)
Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds): Handbook of Semidefinite Programming, volume 27 of International Series in Operations Research and Management Science. Kluwer, Boston (2000)
Zhou B., Cho H.H.: Remarks on spectral radius and Laplacian eigenvalues of a graph. Czech. Math. J. 55(3), 781–790 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Göring, F., Helmberg, C. & Reiss, S. Graph realizations associated with minimizing the maximum eigenvalue of the Laplacian. Math. Program. 131, 95–111 (2012). https://doi.org/10.1007/s10107-010-0344-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-010-0344-z
Keywords
- Spectral graph theory
- Semidefinite programming
- Eigenvalue optimization
- Embedding
- Graph partitioning
- Tree-width