# 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.

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### Existence of connected set with large edge boundary

Let $\Gamma=(V,E)$ be a finite connected graph.
Pretty standard notation. Given a set $S\subset V$, write $\Gamma|_S$ for the restriction of $\Gamma$ to $S$, i.e., the subgraph $(S,\{\{v,w\}\in E: v,w\...

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### Arbitrary-dimensional expanders?

Rephrasing expansion (slightly). Consider the following slightly tweaked version of the usual definition of a (spectral) expander graph.
(We write a weighted graph as $(V,\beta)$, where the weight $\...

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### Vertex percolation on bipartite graphs

This is essentially a repost of this math overflow post since it didn't get any reception.
Let $G = (V \cup C, \mathcal{E})$ be $(\gamma_V, \delta_A, \gamma_B, \delta_B)$-left-right-expanding with ...

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### Construction of graphs of high girth and chromatic number

Are there any concrete constructions of graphs of both high girth and chromatic number?
Of course there is the seminal paper of Erdős which proves the existence of such graphs via the probabilistic ...

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### Can connectivity be less than min cut/degree?

Suppose we have a graph with min-cut $\lambda$ and minimum degree $> \lambda$.
Is it possible for there to be a vertex that is at most $\lambda$-connected to every other vertex in the graph?
...

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### Ramanujan graphs from varieties over finite fields

Let $G$ be a $d$-regular graph. Let $d= \lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n \ge -d $ be the eigenvalues of the adjacency matrix of $G$, and set $\lambda = \max (|\lambda_2| , |\lambda_n|) $...

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### Is there a notion of "rapid" expansion for graphs?

I'll be as intuitive and hand-wavy as possible so as to get to what I am looking for.
Suppose we have a graph $G=(V,E)$. One notion of expansion, namely vertex expansion, can be intuitively described ...

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### An introductory text on expanders

I am looking for a book that covers expander graphs rigorously. Preferably a book aimed at beginners.

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### Spectral norm and "operator norm" for hypergraphs

Consider a $d$-regular, $k$-uniform hypergraph: the elements $S$ of its set $E$ of edges are subsets of $V$ of size $k$, and each vertex $v\in V$ is in $d$ edges. We can then define its adjacency ...

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### Expansion in hypergraphs

Is there a useful concept of expansion in hypergraphs, generalizing the concept for graphs (see: expander graphs)?
Of course, expander graphs can be characterized in several qualitatively equivalent ...

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### (Weakly) connected sets with large (out-)boundary

Let $\Gamma=(V,E)$ be a connected undirected graph with n vertices such that every vertex has degree at least $4$. Now draw arrows on some of the edges, in such a way that the in-degree of every ...

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### Minimum size of regular graph with no short cycles

For $d \geq 3$ (degree) and $r \geq 3$ (radius), say that a $d$-regular (finite, simple, non-oriented) graph $G$ is $r$-almost-tree if it contains no cycle of length $\leq 2 r$: in other words, we ...

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### Expansion of random subgraphs of a bi-regular bipartite graph

Let $G = (L, R, E)$ be a bi-regular bipartite graph, with $|L|=n$ and $|R| = C \cdot n$, where $C$ is a large constant. Let $d$ be its (constant) right-degree.
We know $G$ is a good spectral expander. ...

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### Sums over products over short paths in an expander graph

Let $\Gamma=(V,E)$ be an undirected graph of degree $d$. (Say $d$ is a large constant and the number of vertices $n=|V|$ is much larger.) Let $W_0$ be the space of functions $f:V\to \mathbb{C}$ with ...

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### Examples of 3-transitive expander family of Schreier graphs

What are examples of expander family of 3-transitive Schreier graphs?
Meaning for an action that is 3-transitive.
It is better to have an option for randomization. We know that choosing 2 elements ...

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### Expansion of product of simple Lie group

(a quite technical question if you want to skip).
I am looking at the paper Breuillard, Green, Guralnick, and Tao - Expansion in finite simple groups of Lie type; Specifically, proposition 8.4.
...

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### Reference request: maximal Cheeger constant for 3-regular graphs

Given a connected graph $G$, the Cheeger constant $h(G)$ (a.k.a. Cheeger number or isoperimetric number) roughly measures the "bottleneckedness" of $G$. See Wikipedia for the precise definition.
I ...

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### Expander graphs with many 4-cycles

The question is not strictly well-defined. But it goes like this:
Could you find an infinite family of graphs $G_i$, that are all $\epsilon$-expanders, but have many 4-cycles?
$\epsilon$ should ...

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### Understanding Gillman's proof of the Chernoff bound for expander graphs

My question is about the proof of Claim 1 in this paper: Gillman (1993).
At the end of the proof, the author says:
The matrix product $U^\top\sqrt{D^{-1}}(P+(\mathrm{e}^x-1)B(0)-\mu I)\sqrt{D}U$, ...

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### 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
$\varphi:V\rightarrow\mathbb{C}$ , $\varphi\in L^2(G,\nu)$ ($\nu(v):=\deg(v)$) ...

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### 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) = \...

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### 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 $\...

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### 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 ...

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### 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 $\varepsilon>0$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $X_n$, ...

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### 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-...

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### 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 ...

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### 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 \...

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### 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 $n$, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ``...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\...

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### 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-...

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### 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, ...

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### 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, ...

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### 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 (...

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### 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 ...

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### 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( ...

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### 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.,...