By Andries E. Brouwer, Willem H. Haemers

This ebook offers an straight forward remedy of the fundamental fabric approximately graph spectra, either for usual, and Laplace and Seidel spectra. The textual content progresses systematically, through masking usual themes earlier than featuring a few new fabric on bushes, strongly general graphs, two-graphs, organization schemes, p-ranks of configurations and comparable themes. routines on the finish of every one bankruptcy offer perform and range from effortless but attention-grabbing functions of the taken care of thought, to little tours into similar issues. Tables, references on the finish of the publication, an writer and topic index improve the text.

Spectra of Graphs is written for researchers, lecturers and graduate scholars drawn to graph spectra. The reader is believed to be acquainted with uncomplicated linear algebra and eigenvalues, even if a few extra complex themes in linear algebra, just like the Perron-Frobenius theorem and eigenvalue interlacing are incorporated.

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Deﬁne a matrix C = (ci j ) by ci j = 1 x Ax j . ||x ||2 i i (i) The eigenvalues of C interlace the eigenvalues of A. (ii) If the interlacing is tight, then Ax j = ∑ ci j xi for all j. lies between the smallest and largest (iii) Let x = ∑ x j . The number r := xx Ax x eigenvalue of C. If x is an eigenvector of A with eigenvalue θ , then also C has an eigenvalue θ (for eigenvector 1). Proof Let K be the diagonal matrix with Kii = ||xi ||. Let R be the n × m matrix with columns x j , and put S = RK −1 .

1(iii) we ﬁnd the following. 1 (i) A graph Γ is bipartite if and only if, for each eigenvalue θ of Γ , also −θ is an eigenvalue, with the same multiplicity. (ii) If Γ is connected with largest eigenvalue θ1 , then Γ is bipartite if and only if −θ1 is an eigenvalue of Γ . Proof For connected graphs all is clear from the Perron-Frobenius theorem. That gives (ii) and (by taking unions) the “only if” part of (i). For the “if” part of (i), let θ1 be the spectral radius of Γ . Then some connected component of Γ has eigenvalues θ1 and −θ1 and hence is bipartite.

T. A is deﬁned as u Au . u u Let u1 , . . , un be an orthonormal set of eigenvectors of A, say with Aui = θi ui , where θ1 ≥ . . ≥ θn . If u = ∑ αi ui , then u u = ∑ αi2 and u Au = ∑ αi2 θi . It follows that u Au ≥ θi if u ∈ u1 , . . , ui u u and 26 2 Linear Algebra u Au ≤ θi if u ∈ u1 , . . , ui−1 ⊥ . u u In both cases, equality implies that u is a θi -eigenvector of A. 1 (Courant-Fischer) Let W be an i-subspace of V . Then u Au u∈W, u=0 u u θi ≥ min and θi+1 ≤ Proof max u∈W ⊥ , u=0 u Au .