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...om walk on an undirected graph is determined by the expansion ratio of the graph. We now consider the random walks in a more general setting, and study the ...the transition matrix <math>P</math> instead of the adjacency matrix of a graph. With the same argument as the spectrum of graphs, we can show that <math>\ ...

...The function ''f'' is a bijection if every horizontal line intersects the graph of ''f'' in '''exactly''' one point. ...very horizontal line (in the codomain) intersects exactly one point of the graph. ...

.../Expander Graphs and Mixing|Expander Graphs and Mixing]]: expander graphs, graph spectrum, spectral gap, Cheeger's inequality, rapid mixing of expander walk = The Probability Theory Toolkit = ...

* a directed graph <math>G(V,E)</math>; A fundamental fact in flow theory is that cuts always upper bound flows. ...

* a directed graph <math>G(V,E)</math>; A fundamental fact in flow theory is that cuts always upper bound flows. ...

.../Expander Graphs and Mixing|Expander Graphs and Mixing]]: expander graphs, graph spectrum, spectral gap, Cheeger's inequality, rapid mixing of expander walk = The Probability Theory Toolkit = ...

= Graph Coloring = A '''coloring''' of a graph <math>G(V,E)</math> is a mapping <math>\sigma:V\rightarrow[q]</math> for so ...

.../Expander Graphs and Mixing|Expander Graphs and Mixing]]: expander graphs, graph spectrum, spectral gap, Cheeger's inequality, rapid mixing of expander walk = The Probability Theory Toolkit = ...

...nt is adjacent to the events which are dependent with it in the dependency graph. |Definition (dependency graph)| ...

Formally, a boolean circuit is a directed acyclic graph. Nodes with indegree zero are input nodes, labeled <math>x_1, x_2, \ldots , ...ns that no matter how we color the edges of <math>K_6</math> (the complete graph on six vertices), there must be a '''monochromatic''' <math>K_3</math> (a t ...

== Principles in probability theory == * '''Basics of probability theory''': probability space, events, the union bound, independence, conditional p ...

== Principles in probability theory == * '''Basics of probability theory''': probability space, events, the union bound, independence, conditional p ...

=== Ramsey's theorem for graph === {{Theorem|Ramsey's Theorem (graph, multicolor)| ...

=== Ramsey's theorem for graph === {{Theorem|Ramsey's Theorem (graph, multicolor)| ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...permutation <math>\pi</math> of <math>\{1,\ldots,n\}</math>, define the '''graph''' <math>G_\pi(V,E)</math> as ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...permutation <math>\pi</math> of <math>\{1,\ldots,n\}</math>, define the '''graph''' <math>G_\pi(V,E)</math> as ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...permutation <math>\pi</math> of <math>\{1,\ldots,n\}</math>, define the '''graph''' <math>G_\pi(V,E)</math> as ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...permutation <math>\pi</math> of <math>\{1,\ldots,n\}</math>, define the '''graph''' <math>G_\pi(V,E)</math> as ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...permutation <math>\pi</math> of <math>\{1,\ldots,n\}</math>, define the '''graph''' <math>G_\pi(V,E)</math> as ...

...s]] (like [[light]]) work. It is also called "quantum physics" or "quantum theory". ...e to explain the light that comes from glowing [[hydrogen]]. The quantum [[theory]] of the [[atom]] also had to explain why the [[electron]] stays in its [[o ...