Explicit construction of linear sized tolerant networks
Noga Alon,Fan Chung +1 more
TLDR
This paper constructs explicitly graphs with O(m/@e) vertices and maximum degree O(1/@ e^2), such that after removing any (1-@e%) portion of their vertices or edges, the remaining graph still contains a path of length m.About:
This article is published in Discrete Mathematics.The article was published on 1988-12-01 and is currently open access. It has received 412 citations till now. The article focuses on the topics: Expander mixing lemma.read more
Citations
More filters
Book
The Probabilistic Method
TL;DR: A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.
BookDOI
Spectra of graphs
TL;DR: This book gives an elementary treatment of the basic material about graph Spectra, both for ordinary, and Laplace and Seidel spectra, 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.
Journal ArticleDOI
Expander graphs and their applications
S Hoory,Nathan Linial +1 more
TL;DR: Expander graphs were first defined by Bassalygo and Pinsker in the early 1970s, and their existence was proved in the late 1970s as discussed by the authors and early 1980s.
THE LAPLACIAN SPECTRUM OF GRAPHS y
TL;DR: A survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Lapla-cian eigenvalue 2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidth-type parameters of a graph is given in this article.
Journal ArticleDOI
Expander codes
TL;DR: Using expander graphs, a new family of asymptotically good, linear error-correcting codes are constructed that have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors.
References
More filters
Book
Multiplicative number theory
TL;DR: In this article, the General Modulus is used to describe the distribution of the Primes in arithmetic progression. But the explicit formula for psi(x,chi) is different from the explicit Formula for xi(s) and xi (s,chi).
Book
Combinatorial problems and exercises
TL;DR: In this article, the authors present a dictionary of combinatorial phrases and concepts used in graph theory, including the sieve, the sieving, and the graph sieve.
Journal ArticleDOI
Eigen values and expanders
Noga Alon,Noga Alon +1 more
TL;DR: It is shown that a regular bipartite graph is an expanderif and only if the second largest eigenvalue of its adjacency matrix is well separated from the first.
Journal ArticleDOI
λ1, Isoperimetric inequalities for graphs, and superconcentrators
Noga Alon,Vitali Milman +1 more
TL;DR: This method uses the second smallest eigenvalue of a certain matrix associated with the graph and it is the discrete version of a method used before for Riemannian manifolds for asymptotic isoperimetric inequalities for families of graphs.
Journal ArticleDOI
Hamiltonian circuits in random graphs
TL;DR: The probability that a random graph with n vertices and cn log n edges contains a Hamiltonian circuit tends to 1 as n -> ~ (if c is sufficiently large).