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In mathematics, the Cheeger constant (also Cheeger number or isoperimetric number) of a graph is a numerical measure of whether or not a graph has a "bottleneck". The Cheeger constant as a measure of "bottleneckedness" is of great interest in many areas: for example, constructing well-connected networks of computers, card shuffling.
In 1982, Peter Buser proved a reverse version of this inequality, and the two inequalities put together are sometimes called the Cheeger-Buser inequality. These inequalities were highly influential not only in Riemannian geometry and global analysis , but also in the theory of Markov chains and in graph theory , where they have inspired the ...
For instance, to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < 1 / 2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1 / 2 .
Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.
This can be concisely written as the matrix inequality , where A is an m×n matrix, x is an n×1 column vector of variables, and b is an m×1 column vector of constants. [citation needed] In the above systems both strict and non-strict inequalities may be used. Not all systems of linear inequalities have solutions.
Thus we can find a graph with at least e − cr(G) edges and n vertices with no crossings, and is thus a planar graph. But from Euler's formula we must then have e − cr(G) ≤ 3n, and the claim follows. (In fact we have e − cr(G) ≤ 3n − 6 for n ≥ 3). To obtain the actual crossing number inequality, we now use a probabilistic argument.
Both Plünnecke's proof of Plünnecke's inequality and Ruzsa's original proof of the Plünnecke–Ruzsa inequality use the method of Plünnecke graphs. Plünnecke graphs are a way to capture the additive structure of the sets A , A + B , A + 2 B , … {\displaystyle A,A+B,A+2B,\dots } in a graph theoretic manner [ 5 ] [ 6 ]
The Grothendieck inequality of a graph is an extension of the Grothendieck inequality because the former inequality is the special case of the latter inequality when is a bipartite graph with two copies of {, …,} as its bipartition classes. Thus,