|
|
| Line 1: |
Line 1: |
| The '''[[expander graph|expander]] mixing lemma''' states that, for any two [[subsets]] <math>S, T</math> of a d-regular [[expander graph]] <math>G</math>, the number of edges between <math>S</math> and <math>T</math> is approximately what you would expect in a [[random graph|random]] ''d''-[[regular graph]], i.e. <math>d \cdot|S| \cdot |T| / n</math>.
| | Hi there, I am Alyson Boon even though it is not the title on my beginning certification. My spouse and I reside in Mississippi but now I'm contemplating other options. Credit authorising is how she tends to make a residing. She is really fond of caving but she doesn't have the time recently.<br><br>Here is my blog ... [http://Clothingcarearchworth.com/index.php?document_srl=441551&mid=customer_review clairvoyants] |
| | |
| ==Statement==
| |
| Let <math>G = (V, E)</math> be a d-regular graph with normalized second-largest eigenvalue <math>\lambda</math> (in absolute value) of the adjacency matrix. Then for any two subsets <math>S, T \subseteq V</math>, let <math>E(S, T)</math> denote the number of edges between S and T.
| |
| If the two sets are not disjoint, edges in their intersection are counted twice, that is,
| |
| <math>E(S,T)=2|E(G[S\cap T])| + E(S\setminus T,T) + E(S,T\setminus S)</math>.
| |
| We have
| |
| | |
| :<math>\left|E(S, T) - \frac{d\cdot |S| \cdot |T|}{n}\right| \leq d \lambda \sqrt{|S| \cdot |T|}\,.</math>
| |
| | |
| For a proof, see references.
| |
| | |
| ==Converse==
| |
| Recently, Bilu and [[Nati Linial|Linial]] showed that the converse holds as well: if a graph satisfies the conclusion of the expander mixing lemma, that is, for any two subsets <math>S, T \subseteq V</math>,
| |
| | |
| :<math>|E(S, T) - \frac{d \cdot |S| \cdot |T|}{n}| \leq d \lambda \sqrt{|S| \cdot |T|}</math>
| |
| | |
| then its second-largest eigenvalue is <math>O(d \lambda\cdot (1+\log(1/\lambda)))</math>.
| |
| | |
| ==References==
| |
| *Notes proving the expander mixing lemma. [http://www.tcs.tifr.res.in/~prahladh/teaching/05spring/lectures/lec2.pdf]
| |
| *Expander mixing lemma converse. [http://www.cs.huji.ac.il/~nati/PAPERS/raman_lift.pdf]
| |
| | |
| {{comp-sci-theory-stub}}
| |
| | |
| [[Category:Theoretical computer science]]
| |
| [[Category:Graph theory]]
| |
| [[Category:Lemmas]]
| |
Hi there, I am Alyson Boon even though it is not the title on my beginning certification. My spouse and I reside in Mississippi but now I'm contemplating other options. Credit authorising is how she tends to make a residing. She is really fond of caving but she doesn't have the time recently.
Here is my blog ... clairvoyants