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The Shannon capacity of a graph G is bounded from below by α(G), and from above by ϑ(G). [5] In some cases, ϑ(G) and the Shannon capacity coincide; for instance, for the graph of a pentagon, both are equal to √ 5. However, there exist other graphs for which the Shannon capacity and the Lovász number differ. [6]
Shannon capacity of a graph. If G is an undirected graph, it can be used to define a communications channel in which the symbols are the graph vertices, and two ...
The concept of an error-free capacity awaited Claude Shannon, who built on Hartley's observations about a logarithmic measure of information and Nyquist's observations about the effect of bandwidth limitations. Hartley's rate result can be viewed as the capacity of an errorless M-ary channel of symbols per second. Some authors refer to it as a ...
In information theory, the noisy-channel coding theorem (sometimes Shannon's theorem or Shannon's limit), establishes that for any given degree of noise contamination of a communication channel, it is possible (in theory) to communicate discrete data (digital information) nearly error-free up to a computable maximum rate through the channel.
the mutual information, and the channel capacity of a noisy channel, including the promise of perfect loss-free communication given by the noisy-channel coding theorem; the practical result of the Shannon–Hartley law for the channel capacity of a Gaussian channel; as well as; the bit—a new way of seeing the most fundamental unit of information.
Shannon's diagram of a general communications system, showing the process by which a message sent becomes the message received (possibly corrupted by noise) This work is known for introducing the concepts of channel capacity as well as the noisy channel coding theorem. Shannon's article laid out the basic elements of communication:
The concept of information entropy was introduced by Claude Shannon in his 1948 paper "A Mathematical Theory of Communication", [2] [3] and is also referred to as Shannon entropy. Shannon's theory defines a data communication system composed of three elements: a source of data, a communication channel, and a receiver. The "fundamental problem ...
In graph theory, the Lovász number of a graph is a real number that is an upper bound on the Shannon capacity of the graph. It is also known as Lovász theta function and is commonly denoted by ϑ ( G ) {\displaystyle \vartheta (G)} , using a script form of the Greek letter theta to contrast with the upright theta used for Shannon capacity.