An Introduction to the Theory of Graph Spectra

By: DragoÅ¡ CvetkoviÄ?, Peter Rowlinson, Slobodan SimiÄ?

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Paperback | 15 October 2009

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This introductory text explores the theory of graph spectra: a topic with applications across a wide range of subjects, including computer science, quantum chemistry and electrical engineering. The spectra examined here are those of the adjacency matrix, the Seidel matrix, the Laplacian, the normalized Laplacian and the signless Laplacian of a finite simple graph. The underlying theme of the book is the relation between the eigenvalues and structure of a graph. Designed as an introductory text for graduate students, or anyone using the theory of graph spectra, this self-contained treatment assumes only a little knowledge of graph theory and linear algebra. The authors include many new developments in the field which arise as a result of rapidly expanding interest in the area. Exercises, spectral data and proofs of required results are also provided. The end-of-chapter notes serve as a practical guide to the extensive bibliography of over 500 items.

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Non-Fiction Mathematics Combinatorics & Graph Theory Discrete Mathematics

Preface	p. ix
Introduction	p. 1
Graph spectra	p. 1
Some more graph-theoretic notions	p. 6
Some results from linear algebra	p. 11
Exercises	p. 21
Notes	p. 23
Graph operations and modifications	p. 24
Complement, union and join of graphs	p. 24
Coalescence and related graph compositions	p. 29
General reduction procedures	p. 35
Line graphs and related operations	p. 38
Cartesian type operations	p. 43
Spectra of graphs of particular types	p. 46
Exercises	p. 49
Notes	p. 51
Spectrum and structure	p. 52
Counting certain subgraphs	p. 52
Regularity and bipartiteness	p. 55
Connectedness and metric invariants	p. 58
Line graphs and related graphs	p. 60
More on regular graphs	p. 65
Strongly regular graphs	p. 70
Distance-regular graphs	p. 76
Automorphisms and eigenspaces	p. 80
Equitable partitions, divisors and main eigenvalues	p. 83
Spectral bounds for graph invariants	p. 87
Constraints on individual eigenvalues	p. 91
Exercises	p. 100
Notes	p. 102
Characterizations by spectra	p. 104
Speclial characterizations of certain classes of graphs	p. 104
Cospectral graphs and the graph isomorphism problem	p. 118
Characterizations by eigenvalues and angles	p. 126
Exercises	p. 133
Notes	p. 134
Structure and one eigenvalue	p. 136
Star complements	p. 136
Construction and characterization	p. 141
Bounds on multiplicities	p. 150
Graphs with least eigenvalue-2	p. 154
Graph foundations	p. 155
Exercises	p. 160
Notes	p. 161
Spectral techniques	p. 162
Decompositions of complete graphs	p. 162
Graph homomorphisms	p. 165
The Friendship Theorem	p. 167
Moore graphs	p. 169
Generalized quadrangles	p. 172
Equiangular lines	p. 174
Counting walks	p. 179
Exercises	p. 182
Notes	p. 183
Laplacians	p. 184
The Laplacian spectrum	p. 184
The Matrix-Tree Theorem	p. 189
The largest eigenvalue	p. 193
Algebraic connectivity	p. 197
Laplacian eigenvalues and graph structure	p. 199
Expansion	p. 208
The normalized Laplacian matrix	p. 212
The signless Laplacian	p. 216
Exercises	p. 225
Notes	p. 226
Some additional results	p. 228
More on graph eigenvalues	p. 228
Eigenvectors and structure	p. 243
Reconstructing the characteristic polynomial	p. 250
Integral graphs	p. 254
Exercises	p. 257
Notes	p. 258
Applications	p. 259
Physics	p. 259
Chemistry	p. 266
Computer science	p. 273
Mathematics	p. 277
Notes	p. 283
Appendix	p. 285
The spectra and characteristic polynomials of the adjacency matrix, Seidel matrix, Laplacian and signless Laplacian for connected graphs with at most 5 vertices	p. 286
The eigenvalues, angles and main angles of connected graphs with 2 to 5 vertices	p. 290
The spectra and characteristic polynomials of the adjacency matrix for connected graphs with 6 vertices	p. 294
The spectra and characteristic polynomials of the adjacency matrix for trees with at most 9 vertices	p. 305
The spectra and characteristic polynomials of the adjacency matrix for cubic graphs with at most 12 vertices	p. 316
References	p. 333
Index of symbols	p. 359
Index of terms	p. 361
Table of Contents provided by Ingram. All Rights Reserved.

An Introduction to the Theory of Graph Spectra

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