Abstract
The study of eigenvalues of graphs has a long history. Since the early days, representation theory and number theory have been very useful for examining the spectra of strongly regular graphs with symmetries. In contrast, recent developments in spectral graph theory concern the effectiveness of eigenvalues in studying general (unstructured) graphs. The concepts and techniques, in large part, use essentially geometric methods.(Still, extremal and explicit constructions are mostly algebraic [20].) There has been a significant increase in the interaction between spectral graph theory and many areas of mathematics as well as other disciplines, such as physics, chemistry, communication theory, and computer sciences.
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References
N. Alon,Eigenvalues and expanders, Combinatorica6 (1986) 86–96.
N. Alon, F. R. K. Chung and R. L. Graham,Routing permutations on graphs via matehings, SIAM J. Discrete Math.7 (1994) 513–530.
N. Alon and V. D. Milman, λ1isoperimetrie inequalities for graphs and supereon-centrators, J. Combin. Theory B 38 (1985), 73–88.
L. Babai and M. Szegedy,Local expansion of symmetrical graphs, Combinatorics, Probab. Comput. 1 (1991), 1–12.
J. Cheeger,A lower bound for the smallest eigenvalue of the Laplacian, Problems in Analysis, (R. C. Gunning, ed.), Princeton Univ. Press, Princeton, NJ (1970) 195–199.
F. R. K. Chung,Diameters and eigenvalues, J. Amer. Math. Soc.2 (1989) 187–196.
F. R. K. Chung, Lectures on Spectral graph theory, CBMS Lecture Notes, 1995, AMS Publications, Providence, RI.
F. R. K. Chung, V. Faber, and T. A. Manteuffel,An upper bound on the diameter of a graph from eigenvalues associated with its Laplacian, SIAM J. Discrete Math.7 (1994) 443–457.
F. R. K. Chung and R. L. Graham,Quasi-random set systems, J. Amer. Math. Soc.4 (1991), 151–196.
F. R. K. Chung, R. L. Graham, and R. M. Wilson,Quasi-random graphs, Combi- natorica 9 (1989) 345–362.
F. R. K. Chung, R. L. Graham, and S.-T. Yau,On sampling with Markov chains, random structures and algorithms, to appear.
F. R. K. Chung, A. Grigor’yan, and S.-T. Yau,Eigenvalues and diameters for manifolds and graphs, Adv. in Math., to appear.
F. R. K. Chung and S.-T. Yau,Eigenvalues of graphs and Sobolev inequalities, to appear in Combinatorics, Probab. Comput.
F. R. K. Chung and S.-T. Yau,A Harnack inequality for homogeneous graphs and subgraphs,Communications in Analysis and Geometry2(1994) 628–639.
F. R. K. Chung and S.-T. Yau,The heat kernels for graphs and induced subgraphs, preprint.
F. R. K. Chung and S.-T. Yau,Heat kernel estimates and eigenvalue inequalities for convex subgraphs, preprint.
P. Diaconis and D. Stroock,Geometric bounds for eigenvalues of Markov chains, Ann. Appl. Prob.1 36–61.
M. Jerrum and A. Sinclair,Approximating the permanent, SIAM J. Comput.18 (1989) 1149–1178.
Peter Li and S. T. Yau,On the parabolic kernel of the Schrödinger operator, Acta Mathematica156, (1986) 153–201.
A. Lubotsky, R. Phillips, and P. Sarnak,Ramanujan graphs, Combinatorica8 (1988) 261–278.
G. A. MargulisExplicit constructions of concentrators, Problemy Peredachi Infor- masii9 (1973) 71–80 (English transl. in Problems Inform. Transmission9 (1975) 325–332.
G. A. MargulisExplicit constructions of concentrators, Problemy Peredachi Infor- masii9 (1973) 71-80 (English transl. in Problems Inform. Transmission9 (1975) 325–332.
G. Polya and S. Szego, Isoperimetric inequalities in mathematical physics, Ann. of Math. Stud., no.27, Princeton University Press, Princeton, NJ (1951).
P. Sarnak, Some Applications of Modular Forms, Cambridge University Press, London and New York (1990).
R. M. Tanner,Explicit construction of concentrators from generalized N-gons, SIAM J. Algebraic Discrete Methods5 (1984) 287–294.
L. G. Valiant,The complexity of computing the permanent, Theoret. Comput. Sci. 8 (1979) 189–201.
S.-T. Yau and R. M. Schoen, Differential Geometry, International Press, Cambridge Massachusetts (1994).
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© 1995 Birkhäser Verlag, Basel, Switzerland
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Chung, F.R.K. (1995). Eigenvalues of Graphs. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_128
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DOI: https://doi.org/10.1007/978-3-0348-9078-6_128
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