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

We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values.

Small Ramsey Numbers

This article is intended as a survey, updating earlier surveys in the area. For completeness of the presentation of both particular questions and the general area, it also contains material on closely related topics such as traceable, pancyclic and hamiltonian-connected graphs and digraphs.

/pdf/advances-on-the-hamiltonian-problem-a-survey-35llxwnipo.pdf

Advances on the Hamiltonian Problem – A Survey

Claw-free graphs—a survey

https://tspace.library.utoronto.ca/bitstream/1807/9473/3/chromatic%20number%20of%20the%20square%20of%20a%20planar%20graph.pdf

A bound on the chromatic number of the square of a planar graph

https://ris.utwente.nl/ws/files/6708998/Bauer00not.pdf

Not every 2-tough graph is Hamiltonian

/pdf/paths-and-cycles-in-colored-graphs-cinyc131si.pdf

Paths and cycles in colored graphs

Given a graph G = (V,E) and a (not necessarily proper) edge-coloring of G, we consider the complexity of finding a spanning tree of G with as many different colors as possible, and of finding one with as few different colors as possible. We show that the first problem is equivalent to finding a common independent set of maximum cardinality in two matroids, implying that there is a polynomial algorithm. We use the minimum dominating set problem to show that the second problem is NP-hard.

/pdf/spanning-trees-with-many-or-few-colors-in-edge-colored-8ctblppbhj.pdf

Spanning trees with many or few colors in edge-colored graphs

Living systems are able to achieve prodigious feats of computation with remarkable speed and efficiency (e.g. navigation in a complex environment, object recognition, decision making, and reasoning). Many of these tasks have not been adequately solved using algorithms running on our most powerful computers. When one examines the physical structure of living systems it is very difficult to understand how indeed such systems are able to perform computation. Two things stand out. First, the simplest living systems have a complexity and sophistication that dwarfs manmade technology. Secondly, these living systems were not designed specifically to do computation. In fact, as Richard Dawkins made clear in his book “The Blind Watchmaker” they were not designed at all. Instead they have been subjected to a long process of Darwinian evolution. Why is it that such a blind process can find solutions to these difficult computational tasks, when our best attempts at the principled design of solutions to computational problems are poor by comparison? It is because natural evolution is a bottom-up process that is able to exploit the emergent physical and chemical properties of molecules. It is par excellence a physical exploitation process. Natural evolution uses the enormous complexity of molecular interactions and also the huge parallelism of physical systems. This is the inspiration for the NASCENCE project which aims to emulate Nature and use computer-controlled Darwinian evolution to create sophisticated information processing systems in materials. The benefit of this approach to computation is that we can use the high parallelism and computational complexity of physical components to solve problems that are either computationally intractable or very difficult to model.

The technological drive to produce ever-smaller devices (Moore’s Law) is leading to the construction of machines at the molecular level. However, the basic computational paradigm is still Von Neumann. Molecular and nanoscale electronics concerns building molecule-sized analogues of transistors and assembling them into designs that are facsimiles of today’s solid state circuits. There are fundamental issues with this approach: molecules cannot be controlled and manipulated in the same way as silicon wafers, and it is unclear that macroscopic top-down design methods are appropriate. The NASCENCE project will use computer-controlled manipulation of physical systems, such as networks of nanoparticles, arrays of carbon nanotubes and films of graphene, to evolve them towards doing useful computation and information processing. This is physical computation in a dish.

Nascence project: nanoscale engineering for novel computation using evolution

https://ris.utwente.nl/ws/files/6626340/Broersma02on.pdf

Haitze J. Broersma

Papers

Not every 2-tough graph is Hamiltonian

Paths and cycles in colored graphs

Spanning trees with many or few colors in edge-colored graphs

Nascence project: nanoscale engineering for novel computation using evolution

On some intriguing problems in hamiltonian graph theory: a survey