Laplacian eigenvectors of graphs: Perron-Frobenius and Faber-Krahn type theorems

Lecture Notes in Mathematics

Volumn 1915, Issue , 2007, Pages 1-119

  Bıyıkoğlu, Türker ^a   Leydold, Josef ^b   Stadler, Peter F ^c  

^a ISIK UNIVERSITY   (Turkey)

^b VIENNA UNIVERSITY OF ECONOMICS AND BUSINESS   (Austria)

^c UNIVERSITY OF LEIPZIG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85072879616     PISSN: 00758434     EISSN: 16179692     Source Type: Book Series    
DOI: 10.1007/978-3-540-73510-6_1     Document Type: Chapter

References (178)

1. C. J. Alpert, A. B. Kahng, and S. Yao. Spectral partitioning with multiple eigenvectors. Discrete Appl. Math., 90:3–26, 1999.
2. E. Angel and V. Zissimopoulos. On the classification of NP-complete problems in terms of their correlation coefficient. Discr. Appl. Math., 99:261–277, 2000.
3. V. I. Arnol’d, M. I. Vishik, Y. S. Il’yashenko, A. S. Kalashnikov, V. A. Kondrat’ev, S. N. Kruzhkov, E. M. Landis, V. M. Millionshchikov, O. A. Olejnik, A. F. Filippov, and M. A. Shubin. Some unsolved problems in the theory of differential equations and mathematical physics. Russian Math. Surveys, 44(4):157–171, 1989.
4. B. Aspvall and J. R. Gilbert. Graph coloring using eigenvalue decomposition. SIAM J. Alg. Disc. Meth., 5(4):526–538, 1984.
5. F. M. Atay, T. Bıyıkoğlu, and J. Jost. Synchronization of networks with prescribed degree distribution. IEEE Trans. Circuits Syst. I: Fundamental Theory and Applications, 53(1):92–98, 2006.
6. A. T. Balaban, editor. Chemical Applications of Graph Theory. Academic Press, London, 1984.
7. R. Bapat, S. Kirkland, and S. Pati. The perturbed Laplacian matrix of a graph. Linear Multilinear Algebra, 49(3):219–242, 2001.
8. R. B. Bapat and S. Pati. Algebraic connectivity and the characteristic set of a graph. Linear Multilinear Algebra, 45(2–3):247–273, 1998.
9. S. Barik and S. Pati. On algebraic connectivity and spectral integral variations of graphs. Linear Algebra Appl., 397:209–222, 2005.
10. F. Barioli, S. Fallat, and L. Hogben. A variant on the graph parameters of Colin de Verdière: Implications to the minimum rank of graphs. Elec. J. Lin. Alg., 13:387–404, 2005.
11. J. Barnes, S. Dokov, R. Acevedoa, and A. Solomon. A note on distance matrices yielding elementary landscapes for the TSP. J. Math. Chem., 31:233–235, 2002.
12. J. W. Barnes, B. Dimova, S. P. Dokov, and A. Solomon. The theory of elementary landscapes. Appl. Math. Lett., 16:337–343, 2003.
13. H. Baumgärtel. Analytic Perturbation Theory for Matrices and Operators. Birkhäuser, Basel, 1985.
14. L. W. Beineke, R. J. Wilson, and P. J. Cameron, editors. Topics in Algebraic Graph Theory, volume 102 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, 2004.
15. M. Belkin and P. Niyogi. Laplacian eigenmaps and spectral techniques for embedding and clustering. In Advances in Neural Information Processing Systems 14 (NIPS 2001), pages 585–591, Cambridge, 2002. MIT Press.
16. G. Benkö, C. Flamm, and P. F. Stadler. A graph-based toy model of chemistry. J. Chem. Inf. Comput. Sci., 43:1085–1093, 2003.
17. N. Biggs. Algebraic Graph Theory. Cambridge University Press, Cambridge UK, 2nd edition, 1994.
18. K. Binder and A. P. Young. Spin glasses: Experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys., 58:801–976, 1986.
19. T. Bıyıkoğlu. A discrete nodal domain theorem for trees. Lin. Algebra Appl., 360:197–205, 2003.
20. T. Bıyıkoğlu and J. Leydold. Faber-Krahn type inequalities for trees. J. Comb. Theory B, 2006. to appear.
21. T. Bıyıkoğlu, W. Hordijk, J. Leydold, T. Pisanski, and P. F. Stadler. Graph Laplacians, nodal domains, and hyperplane arrangements. Lin. Algebra Appl., 390:155–174, 2004.
22. T. Bıyıkoğlu, J. Leydold, and P. F. Stadler. Nodal domain theorems and bipartite subgraphs. Elec. J. Lin. Alg., 13:344–351, 2005.
23. A. Björner, L. Lovász, and P. W. Shor. Chip-firing games on graphs. Eur. J. Comb., 12(4):283–291, 1991.
24. D. Bonchev and D. H. Rouvray, editors. Chemical Graph Theory: Introduction and Fundamentals. Abacus/Gordon and Breach, London, 1991.
25. A. J. Bray and M. A. Moore. Metastable states in spin glasses with short-ranged interactions. J. Phys. C, 14:1313–1327, 1981.
26. R. A. Brualdi and B. L. Shader. Matrices of Sign-solvable Linear Systems, volume 116 of Cambridge Tracts in Mathematics. Cambridge Univ. Press, Cambridge, UK, 1995.
27. R. Bürger. The Mathematical Theory of Selection, Recombination, and Mutation. Wiley, Chichester, UK, 2000.
28. A. Chan and C. D. Godsil. Symmetry and eigenvectors. In G. Hahn and G. Sabidussi, editors, Graph Symmetry, volume 497 of NATO ASI Series, Serie C: Mathematical and Physical Sciences, pages 75–106. Kluwer Academic Publishers, 1997.
29. I. Chavel. Eigenvalues in Riemannian Geometry. Academic Press, Orlando Fl., 1984.
30. W.-K. Chen. On vector spaces associated with a graph. SIAM J. Appl. Math., 20:525–529, 1971.
31. S.-Y. Cheng. Eigenfunctions and nodal sets. Comment. Math. Helvetici, 51:43–55, 1976.
32. F. Chung and R. B. Ellis. A chip-firing game and Dirichlet eigenvalues. Discrete Math., 257(2-3):341–355, 2002. Kleitman and combinatorics: a celebration (Cambridge, MA, 1999).
33. F. Chung, L. Lu, and V. Vu. Eigenvalues of random power law graphs. Ann. Comb., 7:21–33, 2003.
34. F. Chung, L. Lu, and V. Vu. Spectra of random graphs with given expected degrees. Proc. Natl. Acad. Sci. USA, 100:6313–6318, 2003.
35. F. R. K. Chung. Spectral Graph Theory, volume 92 of CBMS. American Mathematical Society, 1997.
36. V. Chvátal and P. L. Hammer. Set–packing and threshold graphs. Res. report, Comp. Sci. Dept. Univ. of Waterloo, Ontario CORR 73–21, 1973.
37. B. Codenotti and L. Margara. Local properties of some NP-complete problems. Technical Report TR 92-021, International Computer Science Institute, Berkeley, CA, 1992.
38. Y. Colin de Verdière. Sur un nouvel invariant des graphes et un critère de planarité. J. Comb. Theory, Ser. B, 50:11–21, 1990.
39. Y. Colin de Verdière. Multiplicités des valeurs propres Laplaciens dis-crets et laplaciens continus. Rendiconti di Matematica, 13:433–460, 1993. (French).
40. Y. Colin de Verdière. On a new graph invariant and a criterion for planarity. In Graph Structure Theory, volume 147 of Contemporary Mathematics, pages 137–147. Amercian Mathematical Society, 1993.
41. Y. Colin de Verdière. Spectres de Graphes. Number 4 in Cours Spécialisés. Société Mathématique de France, 1998.
42. D. G. Corneil, Y. Perl, and L. K. Stewart. Cographs: recognition, application and algorithms. Congressus Numerantium, 43:249–258, 1984.
43. D. G. Corneil, Y. Perl, and L. K. Stewart. A linear recognition algorithm for cographs. SIAM J. Comput., 14:926–934, 1985.
44. R. Courant and D. Hilbert. Methods of Mathematical Physics, Vol. 1. Interscience, New York, 1953.
45. D. M. Cvetković, M. Doob, and H. Sachs. Spectra of Graphs – Theory and Applications, volume New York. Academic Press, 1980.
46. D. M. Cvetković, M. Doob, I. Gutman, and A. Torgašev. Recent Results in the Theory of Graph Spectra, volume 36 of Annals of Discrete Mathematics. North Holland, Amsterdam, New York, Oxford, Tokyo, 1988.
47. D. M. Cvetković, P. Rowlinson, and S. Simić. Eigenspaces of Graphs, volume 66 of Encyclopdia of Mathematics and its Applications. Cambrigdge University Press, Cambridge, UK, 1997.
48. E. B. Davies, J. Leydold, and P. F. Stadler. Discrete nodal domain theorems, 2000. arxiv:math.SP/0009120.
49. E. B. Davies, G. M. L. Gladwell, J. Leydold, and P. F. Stadler. Discrete nodal domain theorems. Lin. Algebra Appl., 336:51–60, 2001.
50. V. M. de Oliveira, J. F. Fontanari, and P. F. Stadler. Metastable states in high order short-range spin glasses. J. Phys. A: Math. Gen., 32: 8793–8802, 1999.
51. B. Derrida. Random energy model: Limit of a family of disordered models. Phys. Rev. Lett., 45:79–82, 1980.
52. B. Derrida. Random-energy model: An exactly solvable model of disor-deres systems. Phys. Rev. B, 24:2613–2626, 1981.
53. B. Derrida and E. Gardner. Metastable states of a spin glass chain at 0 temperature. J. Physique, 47:959–965, 1986.
54. L. Devroye. Non-Uniform Random Variate Generation. Springer-Verlag, New-York, 1986.
55. P. Diaconis. Group Representations in Probability and Statistics. Inst. of Math. Stat., Hayward, CA, 1989.
56. R. Diestel. Graph Theory, volume 173 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 2nd edition, 2000.
57. C. Ding and H. Zha. Spectral Clustering, Ordering and Ranking: Statistical Learning with Matrix Factorizations. Springer, New York, 2007. Available July 2007, ISBN: 0-387-30448-7.
58. A. W. M. Dress and D. S. Rumschitzky. Evolution on sequence space and tensor products of representation spaces. Acta Appl. Math., 11: 103–115, 1988.
59. A. M. Duval and V. Reiner. Perron-Frobenius type results and discrete versions of nodal domain theorems. Lin. Algebra Appl., 294:259–268, 1999.
60. H. Edelsbrunner. Algorithms in Combinatorial Geometry, volume 10 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1987.
61. G. Faber. Beweis, daß unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt. Sitz. bayer. Akad. Wiss., pages 169–172, 1923.
62. I. Faria. Permanental roots and the star degree of a graph. Lin. Algebra Appl., 64:255–265, 1985.
63. F. F. Ferreira, J. F. Fontanari, and P. F. Stadler. Landscape statistics of the low autocorrelated binary string problem. J. Phys. A: Math. Gen., 33:8635–8647, 2000.
64. M. Fiedler. Bounds for eigenvalues of doubly stochastic matrices. Linear Algebra Appl., 5:299–310, 1972.
65. M. Fiedler. Algebraic connectivity of graphs. Czechoslovak Math. J., 23:298–305, 1973.
66. M. Fiedler. Eigenvectors of acyclic matrices. Czechoslovak Math. J., 25: 607–618, 1975.
67. M. Fiedler. A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czechoslovak Math. J., 25: 619–633, 1975.
68. M. Fiedler. Laplacian of graphs and algebraic connectivity. In Combinatorics and graph theory, volume 25 of Banach Cent. Publ., pages 57–70. Banach Cent. Publ., Warsaw, 1989.
69. N. J. Fine. The generalized Walsh functions. Trans. Am. Math. Soc., 69:66–77, 1950.
70. M. E. Fisher. On hearing the shape of a drum. J. Comb. Theory, 1: 105–125, 1966.
71. G. S. Fishman. Monte Carlo. Concepts, Algorithms, and Applications. Springer Series in Operations Research. Springer, New York, 1996.
72. W. Fontana, P. F. Stadler, E. G. Bornberg-Bauer, T. Griesmacher, I. L. Hofacker, M. Tacker, P. Tarazona, E. D. Weinberger, and P. Schuster. RNA folding and combinatory landscapes. Phys. Rev. E, 47:2083–2099, 1993.
73. J. J. Forman, P. A. Clemons, S. L. Schreiber, and S. J. Haggarty. SpectralNet — an application for spectral graph analysis and visualization. BMC Bioinformatics, 6:260, 2006. URL http://www. biomedcentral.com/1471-2105/6/260.
74. P. Fowler, T. Pisanski, and J. Shawe-Taylor. Molecular graph eigenvectors for molecular coordinates. In R. Tamassia and I. Tollis, editors, Graph Drawing, volume 894 of Lecture Notes in Computer Science, pages 282–285. Springer-Verlag, Berlin, 1995.
75. J. Friedman. Some geometric aspects of graphs and their eigenfunctions. Duke Math. J., 69(3):487–525, March 1993.
76. J. Friedman and J.-P. Tillich. Calculus on graphs, 2004. arxiv:cs.DM/0408028.
77. J. Friedman and J.-P. Tillich. Wave equations for graphs and the edge-based Laplacian. Pacific J. Math., 216(2):229–266, 2004.
78. F. R. Gantmacher and M. G. Krein. Oscillation matrices and kernels and small vibrations of mechanical systems. AMS Chelsea Publishing, Providence, RI, revised edition, 2002. Translation based on the 1941 Russian original.
79. R. García-Pelayo and P. F. Stadler. Correlation length, isotropy, and meta-stable states. Physica D, 107:240–254, 1997.
80. M. R. Garey and D. S. Johnson. Computers and Intractability. A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, San Francisco, 1979.
81. G. M. L. Gladwell and H. Zhu. Courant’s nodal line theorem and its discrete counterparts. Quart. J. Mech. Appl. Math., 55(1):1–15, 2002.
82. G. M. L. Gladwell and H. Zhu. The Courant-Herrmann conjecture. ZAMM, Z. Angew. Math. Mech., 83(4):275–281, 2003.
83. C. D. Godsil. Algebraic Combinatorics. Chapman & Hall, New York, 1993.
84. C. D. Godsil. Eigenpolytopes of distance-regular graphs. Can. J. Math., 50(4):739–755, 1998.
85. C. D. Godsil and G. F. Royle. Algebraic Graph Theory. Springer, Heidelberg, 2001.
86. C. Gordon, D. L. Webb, and S. Wolpert. One cannot hear the shape of a drum. Bull. Am. Math. Soc., New Ser., 27:134–138, 1992.
87. A. Graovac, D. Plavšić, M. Kaufman, T. Pisanski, and E. C. Kirby. Application of the adjacency matrix eigenvectors method to geometry determination of toroidal carbon molecules. J. Chem. Phys., 113:1925– 1931, 2000.
88. R. Grone, R. Merris, and V. Sunder. The Laplacian spectrum of a graph. SIAM J. Matrix Anal. Appl., 11:218–238, 1990.
89. D. J. Gross and M. Mèzard. The simplest spin glass. Nucl. Phys. B, 240:431–452, 1984.
90. L. K. Grover. Local search and the local structure of NP-complete problems. Oper. Res. Lett., 12:235–243, 1992.
91. S. Guattery and G. L. Miller. On the quality of spectral separators. SIAM J. Matrix Anal. Appl., 19:701–719, 1998.
92. W. Hackbusch. Elliptic Differential Equations: Theory and Numerical Treatment, volume 18 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1992. translated from German by R. Fadiman and P. D. F. Ion.
93. P. L. Hammer and A. K. Kelmans. Laplacian spectra and spanning trees of threshold graphs. Discrete Appl. Math., 65:255–273, 1996.
94. F. Harary. Graph theory. Addison-Wesley, Reading MA, 1969.
95. D. Heidrich, W. Kliesch, and W. Quapp. Properties of Chemically Interesting Potential Energy Surfaces, volume 56 of Lecture Notes in Chemistry. Springer-Verlag, Berlin, 1991.
96. J. Hofbauer and K. Sigmund. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge U.K., 1998.
97. L. Hogben. Spectral graph theory and the inverse eigenvalue problem of a graph. Electron. J. Linear Algebra, 14:12–31, 2005.
98. W. Hordijk and P. F. Stadler. Amplitude spectra of fitness landscapes. Adv. Complex Syst., 1:39–66, 1998.
99. W. Hörmann, J. Leydold, and G. Derflinger. Automatic Nonuniform Random Variate Generation. Springer-Verlag, Berlin Heidelberg, 2004.
100. R. A. Horn and C. R. Johnson. Matrix Analysis. Reprinted with corrections. Cambridge University Press, 1990.
101. E. Hückel. Quantentheoretische Beiträge zum Benzolproblem. Z. Phys., 70:204–286, 1931.
102. W. Imrich and S. Klavžar. Product Graphs: Structure and Recognition. Wiley, New York, 2000.
103. M. Ipsen and A. S. Mikhailov. Evolutionary reconstruction of networks. Phys. Rev. E, 66(4):046109, 2, 2002.
104. F. John. Partial Differential Equations, volume 1 of Applied Mathematical Science. Springer, New York, 1971.
105. J. Jost and M. P. Joy. Spectral properties and synchronization in coupled map lattices. Phys. Rev. E (3), 65:016201, 9, 2002.
106. M. Kac. Can one hear the shape of a drum? Am. Math. Mon., 73(4, Part II):1–23, 1966.
107. A. Katsuda and H. Urakawa. The first eigenvalue of the discrete Dirichlet problem for a graph. J. Comb. Math. Comb. Comput., 27:217–225, 1998.
108. A. Katsuda and H. Urakawa. The Faber-Krahn type isoperimetric inequalities for a graph. Tohoku Math. J., II. Ser., 51(2):267–281, 1999.
109. S. Kirkland. Constructions for type I trees with nonisomorphic Perron branches. Czech. Math. J., 49(3):617–632, 1999.
110. S. Kirkland. An upper bound on algebraic connectivity of graphs with many cutpoints. Electron. J. Linear Algebra, 8:94–109, 2001.
111. S. Kirkland and S. Fallat. Perron components and algebraic connectivity for weighted graphs. Linear Multilinear Algebra, 44(2):131–148, 1998.
112. S. Kirkland, M. Neumann, and B. L. Shader. Characteristic vertices of weighted trees via Perron values. Linear Multilinear Algebra, 40(4): 311–325, 1996.
113. S. J. Kirkland and M. Neumann. On algebraic connectivity as a function of an edge weight. Linear Multilinear Algebra, 52(1):17–23, 2004.
114. S. J. Kirkland, M. Neumann, and B. L. Shader. Bounds on the subdom-inant eigenvalue involving group inverses with applications to graphs. Czech. Math. J., 48(1):1–20, 1998.
115. Y. Koren. Drawing graphs by eigenvectors: theory and practice. Comp. & Math. Appl., 49:1867–1888, 2005.
116. E. Krahn. Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann., 94:97–100, 1925.
117. M. Krivelevich and B. Sudakov. The largest eigenvalue of sparse random graphs. Combin. Probab. Comput., 12(1):61–72, 2003.
118. J. R. Kuttler and V. G. Sigillito. Eigenvalues of the Laplacian in two dimensions. SIAM Review, 26(2):163–193, 1984.
119. I. Lászlo and A. Rassat. The geometric structure of deformed nanotubes and the topological coordinates. J. Chem. Inf. Comput. Sci., 43:519– 524, 2003.
120. I. Lászlo, A. Rassat, P. W. Fowler, and A. Graovac. Topological coordinates for toroidal structures. Chem. Phys. Lett., 342:369–374, 2001.
121. H. Lewy. On the minimum number of domains in which the nodal lines of spherical harmonics divide the sphere. Comm. partial Diff. Eqns., 2: 1233–1244, 1977.
122. J. Leydold. On the number of nodal domains of spherical harmonics. Topology, 35:301–321, 1996.
123. J. Leydold. A Faber-Krahn-type inequality for regular trees. GAFA, Geom. Funct. Anal., 7(2):364–378, 1997.
124. J. Leydold. The geometry of regular trees with the Faber-Krahn property. Discrete Math., 245(1–3):155–172, 2002.
125. Y. Li and L. Guan. Some remarks for discrete versions of nodal domain theorems. J. Math. Res. Exposition, 23(2):275–278, 2003.
126. L. Lovász. Spectra of graphs with transitive groups. Periodica Math. Hung., 6:191–195, 1975.
127. L. Lovász and A. Schrijver. A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs. Proc. Amer. Math. Soc., 126:1275–1285, 1998.
128. L. Lovász and A. Schrijver. On the null space of a Colin de Verdière matrix. Ann. Inst. Fourier, 49(3):1017–1026, 1999.
129. S. Maslov. Measures of globalization based on cross-correlations of world financial indices. Physica A, 301:397–406, 2001.
130. R. Merris. Characteristic vertices of trees. Linear Multilinear Algebra, 22(2):115–131, 1987.
131. R. Merris. Laplacian matrices of graphs: A survey. Lin. Algebra Appl., 197–198:143–176, 1994.
132. R. Merris. Laplacian graph eigenvectors. Lin. Algebra Appl., 278:221– 236, 1998.
133. M. Mézard, G. Parisi, and M. Virasoro. Spin Glass Theory and Beyond. World Scientific, Singapore, 1987.
134. P. G. Mezey. Potential Energy Hypersurfaces. Elsevier, Amsterdam, 1987.
135. B. Mohar. The Laplacian spectrum of graphs. In Y. Alavi, G. Char-trand, O. Ollermann, and A. Schwenk, editors, Graph Theory, Combinatorics, and Applications, pages 871–898, New York, 1991. John Wiley and Sons, Inc.
136. B. Mohar. Some applications of Laplace eigenvalues of graphs. In G. Hahn and G. Sabidussi, editors, Graph Symmetry: Algebraic Methods and Applications, volume 497 of NATO ASI Series C: Mathematical and Physical Sciences, pages 225–275. Kluwer Academic Publishers, 1997.
137. B. Mohar and S. Poljak. Eigenvalues in combinatorial optimization. In R. A. Brualdi, S. Friedland, and V. Klee, editors, Combinatorial and Graph-Theoretical Problems in Linear Algebra, volume 50 of IMA Volumes in Mathematics and Its Applications, pages 107–151, Berlin, 1993. Springer-Verlag.
138. J. J. Molitierno. The spectral radius of submatrices of Laplacian matrices for trees and its comparison to the Fiedler vector. Linear Algebra Appl., 406:253–271, 2005.
139. H. Müller. private communication, 2001.
140. R. Palmer. Optimization on rugged landscapes. In A. S. Perelson and S. A. Kauffman, editors, Molecular Evolution on Rugged Landscapes: Proteins, RNA, and the Immune System, pages 3–25. Addison Wesley, Redwood City, CA, 1991.
141. S. Pati. The third smallest eigenvalue of the Laplacian matrix. Electron. J. Linear Algebra, 8:128–139, 2001.
142. T. Pisanski and J. Shawe-Taylor. Characterising graph drawing with eigenvectors. J. Chem. Inf. Comput. Sci., 40:567–571, 2000.
143. T. Pisanski and A. Žitnik. Representations of graphs and maps. In J. Gross and T. Tucker, editors, Topological Graph Theory, Encyclopedia of Mathematics. Cambridge University Press, 2005. in preparation.
144. A. Pothen, H. D. Simon, and K.-P. Liou. Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal. Appl., 11:430–452, 1990.
145. D. L. Powers. Graph partitioning by eigenvectors. Lin. Algebra Appl., 101:121–133, 1988.
146. A. R. Pruss. Discrete convolution-rearrangement inequalities and the Faber-Krahn inequality on regular trees. Duke Math. J., 91(3):463–514, 1998.
147. C. M. Reidys and P. F. Stadler. Combinatorial landscapes. SIAM Review, 44:3–54, 2002.
148. H. Rieger. The number of solutions of the Thouless-Anderson-Palmer equations for p-spin interaction spin glasses. Phys. Rev. B, 46:14655– 14661, 1992.
149. D. Rockmore, P. Kostelec, W. Hordijk, and P. F. Stadler. Fast Fourier transform for fitness landscapes. Appl. Comput. Harmonic Anal., 12: 57–76, 2002.
150. R. Roth. On the eigenvectors belonging to the minimum eigenvalue of an essentially nonnegative symmetric matrix with bipartite graph. Lin. Algebra Appl., 118:1–10, 1989.
151. P. Rowlinson. Graph perturbations. In Surveys in Combinatorics, Proc. 13th Br. Comb. Conf. Guildford/UK 1991. Lond. Math. Soc., 1991.
152. J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Trans. Pattern Analysis Machine Intel., 22:888–905, 2000.
153. A. Solomon, J. W. Barnes, S. P. Dokov, and R. Acevedo. Weakly symmetric graphs, elementary landscapes, and the TSP. Appl. Math. Lett., 16:401–407, 2003.
154. B. M. R. Stadler and P. F. Stadler. Generalized topological spaces in evolutionary theory and combinatorial chemistry. J. Chem. Inf. Comput. Sci., 42:577–585, 2002. Proceedings MCC 2001, Dubrovnik.
155. P. F. Stadler. Landscapes and their correlation functions. J. Math. Chem., 20:1–45, 1996.
156. P. F. Stadler. Spectral landscape theory. In J. P. Crutchfield and P. Schuster, editors, Evolutionary Dynamics—Exploring the Interplay of Selection, Neutrality, Accident, and Function, pages 231–272. Oxford University Press, 2002.
157. P. F. Stadler. Canonical approximation of landscapes. Technical Report SFI 94-09-51, Santa Fe Institute, Santa Fe, NM, USA, 1994.
158. P. F. Stadler and R. Happel. Random field models for fitness landscapes. J. Math. Biol., 38:435–478, 1999.
159. P. F. Stadler and R. Happel. Correlation structure of the landscape of the graph-bipartitioning-problem. J. Phys. A: Math. Gen., 25:3103– 3110, 1992.
160. P. F. Stadler and B. Krakhofer. Local minima of p-spin models. Rev. Mex. Fis., 42:355–363, 1996.
161. P. F. Stadler and W. Schnabl. The landscape of the travelling salesman problem. Phys. Lett. A, 161:337–344, 1992.
162. P. F. Stadler, R. Seitz, and G. P. Wagner. Evolvability of complex characters: Population dependent Fourier decomposition of fitness landscapes over recombination spaces. Bull. Math. Biol., 62:399–428, 2000.
163. M. W. Strickberger. Evolution. Jones and Bartlett, Boston MA, 1990.
164. F. Tanaka and S. F. Edwards. Analytic theory of the ground state properties of a spin glass: I. Ising spin glass. J. Phys. F: Metal Phys., 10:2769–2778, 1980.
165. D. J. Thouless, P. W. Anderson, and R. G. Palmer. Solution of ‘Solvable model of a spin glass’. Phil. Mag., 35:593–601, 1977.
166. N. Trinajstić. Chemical Graph Theory. CRC Press, Boca Raton, FL, 2nd edition, 1992.
167. H. van der Holst. Topological and Spectral Graph Characterizations. PhD thesis, Universiteit van Amsterdam, 1996.
168. H. van der Holst, L. Lovász, and S. A. The Colin de Verdière graph parameter. In Graph Theory and Computational Biology (Balatonlelle, 1996), Mathematical Sudies, pages 29–85. Janos Bolyai Math. Soc., Budapest, 1999.
169. J. L. Walsh. A closed set of normal orthogonal functions. Amer. J. Math., 45:5–24, 1923.
170. J. Wang and M. Xu. Quasi-Abelian Cayley graphs and Parsons graphs. Eur. J. Comb., 18:597–600, 1997.
171. E. D. Weinberger. Correlated and uncorrelated fitness landscapes and how to tell the difference. Biol. Cybern., 63:325–336, 1990.
172. E. D. Weinberger. Local properties of Kauffman’s N-k model: A tunably rugged energy landscape. Phys. Rev. A, 44:6399–6413, 1991.
173. D. B. West. Introduction to Graph Theory. Prentice Hall, New York, 1996.
174. H. Whitney. Congruent graphs and the connectivity of graphs. Amer. J. Math., 54:150–168, 1932.
175. S. Wright. The roles of mutation, inbreeding, crossbreeeding and selection in evolution. In D. F. Jones, editor, Proceedings of the Sixth International Congress on Genetics, volume 1, pages 356–366, New York, 1932. Brooklyn Botanic Gardens.
176. S. Wright. “Surfaces” of selective value. Proc. Nat. Acad. Sci. USA, 58: 165–172, 1967.
177. B. Zgrablic. On adjacency-transitive graphs. Discrete Math., 182:321– 332, 1998.
178. G. M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.