Spectra of graphs.

*(English)*Zbl 1231.05001
Universitext. Berlin: Springer (ISBN 978-1-4614-1938-9/hbk; 978-1-4614-1939-6/ebook). xiii, 250 p. (2012).

This book contains an extensive overview of current topics and recent developments in algebraic graph theory, and has a survey-like appearance. It is aimed primarily at researchers and graduate-level students, as it is based on lecture notes for the course that the authors gave at the Institute for Studies in Theoretical Physics and Mathematics in Tehran in 2006.

The choice of topics reflects the authors’ expertise in association schemes and distance-regular graphs, and the book can be roughly divided into two parts.

The first part deals with the theory and applications of eigenvalues and eigenvectors of the adjacency, Laplacian and Seidel matrices of graphs. The important applications of the largest, the second largest and the smallest eigenvalue, as well as interlacing, are covered here, including relations between vertex ranking and the Perron-Frobenius eigenvector, the second largest eigenvalue and expansion properties, graph bisections and clustering and graph drawing. It also contains many theoretical applications of spectral graph properties, such as the proof of the Grone-Merris conjecture, the spectral proof that the Petersen graph is not Hamiltonian, impossibility of decomposition of \(K_{10}\) into three Petersen graphs, lower bound on the number of complete bipartite subgraphs into which a complete graph may be decomposed, as well as the bounds for the independence number, chromatic number and Shannon capacity.

The second half of the book is devoted to the interplay between graph spectra and association schemes and coherent configurations. It covers topics such as root lattices, generalized quadrangles, discussion of (81,20,1,6)-srg, Delsarte’s linear programming bound, as well as recent major developments in distance-regular graphs: a proof of the Bannai-Ito conjecture, Van Dam and Koolen’s construction of twisted Grassmann graphs, and the determination of the connectivity of distance-regular graphs.

Chapter list: 1. Graph spectrum; 2. Linear algebra; 3. Eigenvalues and eigenvectors of graphs; 4. The second-largest eigenvalue; 5. Trees; 6. Groups and graphs; 7. Topology; 8. Euclidean representations; 9. Strongly regular graphs; 10. Regular two-graphs; 11. Association schemes; 12. Distance-regular graphs; 13. \(p\)-ranks; 14. Spectral characterizations; 15. Graphs with few eigenvalues.

The choice of topics reflects the authors’ expertise in association schemes and distance-regular graphs, and the book can be roughly divided into two parts.

The first part deals with the theory and applications of eigenvalues and eigenvectors of the adjacency, Laplacian and Seidel matrices of graphs. The important applications of the largest, the second largest and the smallest eigenvalue, as well as interlacing, are covered here, including relations between vertex ranking and the Perron-Frobenius eigenvector, the second largest eigenvalue and expansion properties, graph bisections and clustering and graph drawing. It also contains many theoretical applications of spectral graph properties, such as the proof of the Grone-Merris conjecture, the spectral proof that the Petersen graph is not Hamiltonian, impossibility of decomposition of \(K_{10}\) into three Petersen graphs, lower bound on the number of complete bipartite subgraphs into which a complete graph may be decomposed, as well as the bounds for the independence number, chromatic number and Shannon capacity.

The second half of the book is devoted to the interplay between graph spectra and association schemes and coherent configurations. It covers topics such as root lattices, generalized quadrangles, discussion of (81,20,1,6)-srg, Delsarte’s linear programming bound, as well as recent major developments in distance-regular graphs: a proof of the Bannai-Ito conjecture, Van Dam and Koolen’s construction of twisted Grassmann graphs, and the determination of the connectivity of distance-regular graphs.

Chapter list: 1. Graph spectrum; 2. Linear algebra; 3. Eigenvalues and eigenvectors of graphs; 4. The second-largest eigenvalue; 5. Trees; 6. Groups and graphs; 7. Topology; 8. Euclidean representations; 9. Strongly regular graphs; 10. Regular two-graphs; 11. Association schemes; 12. Distance-regular graphs; 13. \(p\)-ranks; 14. Spectral characterizations; 15. Graphs with few eigenvalues.

Reviewer: Dragan Stevanović (Niš)

##### MSC:

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |

05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |

05E30 | Association schemes, strongly regular graphs |