This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics. Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra. The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.

Spectra of graphs

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Equiangular lines in Euclidean spaces

On the distance spectrum of graphs

We consider nonregular graphs having precisely three distinct eigenvalues. The focus is mainly on the case of graphs having two distinct valencies and our results include constructions of new examples, structure theorems, valency constraints, and a classification of certain special families of such graphs. We also present a new example of a graph with three valencies and three eigenvalues of which there are currently only finitely many known examples.

Biregular graphs with three eigenvalues

We investigate graphs whose signless Laplacian matrix has three distinct eigenvalues. We show that the largest signless Laplacian eigenvalue of a connected graph G with three distinct signless Laplacian eigenvalues is noninteger if and only if G = K n  − e for n ≥ 4, where K n  − e is the n vertex complete graph with an edge removed. Moreover, examples of such graphs are given in this article.

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A note on graphs whose signless Laplacian has three distinct eigenvalues

https://people.math.sc.edu/lu/teaching/2009spring_778S/adj3eig.pdf

H. Chuang

Papers

Graphs with three distinct eigenvalues and largest eigenvalue less than 8