We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

The theory of directed graphs has developed enormously over recent decades, yet this book (first published in 2000) remains the only book to cover more than a small fraction of the results. New research in the field has made a second edition a necessity. Substantially revised, reorganised and updated, the book now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems. As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem. Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject. Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science. It will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.

Digraphs Theory Algorithms And Applications

Graph minors. II: Algorithmic aspects of tree-width

∑11-Formulae on finite structures

WHAT is the theory of functions about? This question may be heard now and again from a mathematical student; and if, by way of a pattial reply, it be said that the elements of the theory of functions forms the basis on which the whole of that part of pure mathematics which deals with continuously varying quantity rests, the answer would not be too wide nor would it always imply too much. Theory of Functions of a Complex Variable. By Dr. A. R. Forsyth. (Cambridge University Press, 1893.)

Theory of Functions of a Complex Variable

Soit f un entier positif fixe et σ une phrase dans le calcul de predicats d'ordre 1 des fonctions unaires. Pour des entiers positifs n, une n-structure est un modele d'univers {0,1,...n-1} et de f fonctions unaires et μ(nσ) est le rapport du nombre de n structures satisfaisant σ a n nf , le nombre de n-structures. On montre que lim n→∞ μ(nσ) exite pour tout tel σ, et sa valeur est donnee par une expression constituee de constantes entieres et des operateurs +,−,., /, et exp(x)

/pdf/probabilities-of-first-order-sentences-about-unary-functions-4t04pn5059.pdf

Probabilities of first-order sentences about unary functions

For every κ ∈ ω, there is an infinite set A κ ⊆ ω and a d (κ) ∈ ω such that for all Q 0 , Q 1 , ⊆ A κ where ∣ Q 0 ∣ = ∣ Q 1 ∣, or d (κ) Q 0 ∣, Q 1 ∣ 0 , the structures ‹ω, +, Q 0 › and ‹ω, +, Q 1 › are indistinguishable by first-order sentences of quantifier depth κ whose atomic formulas are of the form u = v, u + v = w , and Q ( u ), where u, v , and w are variables.

On Sets of Relations Definable by Addition

Analysis and application of adaptive sampling

The author considers random graphs with edge probability beta n/sup - alpha /, where n is the number of vertices of the graph, beta >0 is fixed, and alpha =1 or alpha =(l+1)/l for some fixed positive integer l. It is proved that, for every first-order sentence, the probability that the sentence is true for the random graph has an asymptotic limit. Also, there is an effective procedure for generating the value of the limit in closed form. >

Probabilities of sentences about very sparse random graphs

A random boolean cellular automaton is a network of boolean gates where the inputs, the boolean function, and the initial state of each gate are chosen randomly. In this article, each gate has two inputs. Let $a$ (respectively $c$) be the probability the the gate is assigned a constant function (respectively a non-canalyzing function, i.e., {\sc equivalence} or {\sc exclusive or}). Previous work has shown that when $a > c$, with probability asymptotic to 1, the random automaton exhibits very stable behavior: almost all of the gates stabilize, almost all of them are weak, i.e., they can be perturbed without affecting the state cycle that is entered, and the state cycle is bounded in size. This article gives evidence that the condition $a = c$ is a threshold of chaotic behavior: with probability asymptotic to 1, almost all of the gates are still stable and weak, but the state cycle size is unbounded. In fact, the average state cycle size is superpolynomial in the number of gates.

http://export.arxiv.org/pdf/adap-org/9309001

On the Threshold of Chaos in Random Boolean Cellular Automata

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

James F. Lynch

Author Tools

Chat about Author