Graphs.- Groups.- Transitive Graphs.- Arc-Transitive Graphs.- Generalized Polygons and Moore Graphs.- Homomorphisms.- Kneser Graphs.- Matrix Theory.- Interlacing.- Strongly Regular Graphs.- Two-Graphs.- Line Graphs and Eigenvalues.- The Laplacian of a Graph.- Cuts and Flows.- The Rank Polynomial.- Knots.- Knots and Eulerian Cycles.- Glossary of Symbols.- Index.

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Algebraic Graph Theory

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

Distance-Regular Graphs

Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

Laplacian energy of a graph

Maximal Energy Graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,  Distance-Regular Graphs , Springer-Verlag, Berlin, 1989] was written.

Distance-regular graphs

The hyperbolicity $ \delta^* \ge 0 $ of a metric space in Gromov's sense can be viewed as a measure of how "tree-like" the space is, since those spaces for which $ \delta^* = 0 $ holds are precisely the set of (metric) trees. Here, we show that any chordal graph equipped with the usual graph metric is in this sense reasonably tree-like. In particular, we prove that the hyperbolicity of any chordal graph is bounded, and is at most two. Moreover, we characterize those chordal graphs with hyperbolicity one.

On the Hyperbolicity of Chordal Graphs

Given a graph $G$, its energy $E(G)$ is defined to be the sum of the absolute values of the eigenvalues of $G$. This quantity is used in chemistry to approximate the total $\pi$-electron energy of molecules and in particular, in case $G$ is bipartite, alternant hydrocarbons. Here we show that if $G$ is a bipartite graph with $n$ vertices, then $E(G) \leq \frac{n}{\sqrt{8}}(\sqrt{2} + \sqrt{n})$ must hold, characterize those bipartite graphs for which this bound is sharp, and provide an infinite family of maximal energy bipartite graphs.

Maximal Energy Bipartite Graphs

We consider a distance-regular graph Γ with diameter i>d ≥ 3 and eigenvalues i>k e t0 > t1 > … > ti>d. We show the intersection numbers i>a1, i>b1 satisfy We say Γ is i>tight whenever Γ is not bipartite, and equality holds above. We characterize the tight property in a number of ways. For example, we show Γ is tight if and only if the intersection numbers are given by certain rational expressions involving i>d independent parameters. We show Γ is tight if and only if i>a1 ≠ 0, i>ad e 0, and Γ is 1-homogeneous in the sense of Nomura. We show Γ is tight if and only if each local graph is connected strongly-regular, with nontrivial eigenvalues −1 − i>b1(1 + t1)−1 and −1 − i>b1(1 + ti>d)−1. Three infinite families and nine sporadic examples of tight distance-regular graphs are given.

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Jack H. Koolen

Papers

Maximal Energy Graphs

Distance-regular graphs

On the Hyperbolicity of Chordal Graphs

Maximal Energy Bipartite Graphs

Tight Distance-Regular Graphs