Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [expander-graphs]

An expander graph is a graph in which small sets of vertices have large 'boundary'. Ramanujan graphs are examples of expanders.

2
votes
1answer
110 views

Spectral radius of Markov averaging operator on graphs

The definition of Markov operator which I am familiar with: For a graph $G=(V,E)$, Markov's operator upon a function $\varphi:V\rightarrow\mathbb{C}$ , $\varphi\in L^2(G,\nu)$ ($\nu(v):=\deg(v)$) ...
1
vote
0answers
65 views

Expander mixing lemma in combinatoric expanders

There is a well-known relation between combinatoric expansion and the gap between the first and the second largest eigenvalues (Dodziuk 1984) : $$ h(G) \le d \sqrt{2 (1 - \alpha)} $$ where $$h(G) = \...
1
vote
0answers
36 views

Tight upper and lower bounds for unbalanced left-regular expander graphs

I am interested in finding the best expansion parameters for unbalanced left-regular expander graphs. Specifically, fix $\delta\in(0,1/2)$, and a positive integer $d$. Let us call a bipartite graph $\...
0
votes
1answer
75 views

Proof of reduction from random walks to martingales - why $T\le k$?

I'm trying to understand the proof of theorem 1.6 from the paper "A Matrix Expander Chernoff Bound". In the proof they say: "Iterating this construction on the remainder a total of $T ≤ k$ times" and ...
2
votes
2answers
206 views

Random walk and isoperimetric constant

I assume that a result of the following kind is known, and I would really appreciate a reference for it... Or at least, some hints as to where to start looking. Theorem(?): Let $\varepsilon>0$ ...
1
vote
0answers
83 views

Short cycles in expander graphs

One often reads that expander graphs look locally like trees. In fact, there are many classes of expander graphs like random regular graphs or some Ramanujan graphs for which it has been shown that ...
-4
votes
2answers
202 views

Is a single randomly generated graph sufficient to prove an almost all colorability result?

I have generated a single random 17th degree 100 vertex graph, with self-loops and multiple edges rerandomized out of existence, so the graph is highly 17 regular, and after long computation with a ...
8
votes
3answers
341 views

Where does $2\sqrt{d-1}$ come from in Ramanujan graphs?

Ramanujan graphs are the best spectral expanders: $\lambda_2 \le 2\sqrt{d-1}$. I'm looking for some intuition for this value $2\sqrt{d-1}$. Friedman showed that every random $d$-regular graph ...
1
vote
1answer
275 views

How should one define expansion for irregular graphs?

This question is related to a previous question: Does it make sense to talk about expansion in irregular graphs? A family of expanders can be defined as a sequence of graphs whose spectral gap is ...
8
votes
1answer
178 views

Does it make sense to talk about expansion in irregular graphs?

Usually, one defines an expander graph to be a regular graph satisfying one of the following properties: Either the edge-expansion is large, or the spectral gap is large, or the mixing time is at most ...
3
votes
0answers
74 views

Analogues of relative property $(\tau)$ for Schreier graphs

Suppose I have an expanding family of Schreier graphs $Z_n=\text{Sch}(G_n,S_n,X_n)$ of groups $G_n=\underbrace{G\wr\ldots\wr G}_{\text{$n$ times}}$ acting on sets $S_n=S^n$ by generating sets $X_n$, ...
4
votes
0answers
69 views

Group of Lie type as expanders: explicit estimates

In a paper Finite simple groups as expanders by M. Kassabov, A. Lubotzky and N. Nikolov there is a theorem, stating that there exists $\varepsilon>0$ and $k\in\mathbb{N}$, such that for every non-...
4
votes
2answers
245 views

Non-Cayley expander graphs

When I search about expander graphs in google I see a lot of articles about expander Cayley graphs. Now my questions are as follows: Are all expander regular graphs are Cayley, or there is a special ...
9
votes
2answers
397 views

Groups without property (T) but all finite quotients are expanders

What is an example of a group $G$ which 1- is finitely generated by $S$, 2- does not have property (T), 3- admits infinitely many finite quotients which do not factor through an homomorphism $G \...
8
votes
1answer
289 views

Cheeger Numbers for 3-regular Graphs

A student wanted a challenging Graph Theory programming project and I had him try to determine the maximum value of the Cheeger number (isoperimetric number) among all 3-regular graphs of order $n$, ...
2
votes
2answers
179 views

How does one calculate/estimate/guarantee the girth of a non-Abelian Cayley graph?

This question is in reference to this other question, Can someone point out references (or explain!) which give techniques of being able to prove for any Cayley graph this property of having a girth ...
1
vote
0answers
122 views

Probabilistic proof for expander existence [closed]

I am new to probabilistic proofs and trying to understand them better. Apparently, a common probabilistic proof focuses on the existence of expanders (eg. vertex expanders). I've been using the search ...
2
votes
1answer
107 views

How many edges guarantee an expander?

Complete graphs are good expanders. Deleting few edges from a complete graph leaves a good expander. What's the proportion of edges that one needs to remove to make the graph a bad expander? Or, is ...
0
votes
1answer
135 views

A particular argument in the review on expanders by Hoory-Linial-Wigderson

I am thinking about the third bullet point on page 455 here, http://www.ams.org/journals/bull/2006-43-04/S0273-0979-06-01126-8/ Can someone explain what is the argument there which seems to conclude ...
7
votes
2answers
398 views

Constructing Ramanujan graphs from elliptic curves

Is there an exposition which explains how to do this step-by-step? (I see stray references and allusions to such a thing being possible but can't locate anything concretely) Something to do with ``...
7
votes
1answer
703 views

When are (Abelian) Cayley graphs also expanders?

I want to ask the question in two parts, (1) Is there some fundamental distinguishing property between Abelian and non-Abelian Cayley graphs? (say some specific proof technique which distinguishes ...
3
votes
2answers
273 views

Matching polynomials and Ramanujan graphs

Is it purely coincidental that the same number $2\sqrt{d-1}$ appears in these two following apparently disparate concepts? A $d-$regular graph is said to be called Ramanujan if its adjacency ...
4
votes
0answers
384 views

The Bilu-Linial conjecture and Ramanujan graphs

The Bilu-Linial conjecture claims that every $d-$regular graph has a $2-$lift such that for the signing matrix has its eigenvalues between $[-2\sqrt{d-1},2\sqrt{d-1}]$ (the ``signing matrix" is the ...
3
votes
0answers
124 views

Are there two-sided $\varepsilon$-expanders with independent sets of size $(1-\varepsilon)n$?

Terry Tao's notes on expander graphs has the following exercise: Exercise 13 Let $G$ be a $k$-regular graph on $n$ vertices that is a two-sided $\epsilon$-expander for some $n > k \geq 1$ and $\...
2
votes
0answers
129 views

Expanding graphs from iterated zig-zag product

Let $\Gamma \, zz\, G$ denote the zig-zag product of graphs $\Gamma$ and $G$. Reingold, Vadhan, and Wigderson proved that the second largest eigenvalue of the normalized adjacency matrix of the zig-...
2
votes
0answers
74 views

Asymptotic results in unbalanced left $d$-regular expander graphs

Let $U = [n]$ and $V = [m]$ be sets of nodes with $n > m$ and $E = U\times V$ be a set of edges. Let $\mathcal{N}(S)$ be the set of neighbors of a subset $S$ from $U$ or $V$. Call a graph $G = (U, ...
10
votes
0answers
546 views

Connections between Riemann hypothesis for curves over finite fields and Ramanujan property for graphs

this question relates to the beautiful construction of expander graphs using Cayley graphs of $PGL_2(\mathbb{F}_q)$, as exposited by Davidoff-Sarnak-Valette in their book, Elementary Number Theory, ...
15
votes
3answers
550 views

Embedding expanders in CAT(0) spaces

It is well-known that expanders are hard to embed into Hilbert (or $\ell^p$) spaces - any embedding of an expander with $n$ vertices has distortion $\Omega(\log n)$. Can anyone provide a reference (...
4
votes
0answers
197 views

What are the best bounds to date on the maximum girth of a cubic graph?

The girth of a graph is the length of its smallest cycle. Studying the maximum possible girth for a $k$-regular graph on $n$ vertices is a very well-studied problem. In the 1988 paper "Ramanujan ...
7
votes
1answer
221 views

Do there exist “expanding” $1$-skeletons of simple $4$-polytopes?

Let $\{ G_n \}_{n \ge 1}$ be a sequence of graphs such that the number of vertices of $G_n$ tends to $\infty$ as $n \to \infty$. We say that $\{ G_n \}_{n \ge 1}$ is an expander family if $\lambda_2( ...
11
votes
1answer
289 views

Variant of an Expander graph: Probability that S random points cast a shadow/projection of size at most S/2 on each face of a cube.

Consider an integer cube of size $\sqrt{k} \times \sqrt{k} \times \sqrt{k}$, where $k$ is an asymptotically large perfect square number. Place k points in this cube at uniformly random locations, i.e.,...
5
votes
1answer
421 views

More expanders?

Having received several exhausting answers to my recent question about the expansion properties of a certain graph, I now wonder whether anything is known on the following graphs of a similar nature: ...
12
votes
2answers
884 views

An expander (?) graph

For a prime $p$, consider the graph on the vertex set ${\mathbb F}_p$, in which every vertex $z$ is adjacent to $z\pm 1$ and also to $z^{-1}$ (unless $z=0$). I was told that this graph is known to be ...
7
votes
1answer
400 views

Constructing expanders in Z/pZ

Fix a positive integer $k>0$. For $p>k$ a prime, let $A_p$ be a subset of the finite field $\mathbb{Z}/p\mathbb{Z}$ of size $k$ which contains a primitive element. Define $G_p$ to be the (di)...
20
votes
0answers
838 views

Could unramified Galois groups satisfy a version of property tau?

This is an experiment: there is a question I want to mention in an article I'm writing, and I am not sure it's a sensible question, so I will ask it here first, in the hopes that if it's insensible ...