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1answer
40 views

References to learn more about graph laplacian.

I have vaguely heard of this connection between random matrix theory and graphs (the spectral gap of their laplacians) on compact Riemann surfaces. Can someone give a pedagogic reference which ...
0
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0answers
77 views

introduction to spectral geometry

In the paper Belkin, Mikhail, and Partha Niyogi. "Laplacian eigenmaps and spectral techniques for embedding and clustering." In NIPS, vol. 14, pp. 585-591. 2001 The ...
1
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0answers
75 views

Constructing a digraph from its spectrum

This is related to the following question which has already been explored: Reverse Graph Spectra Problem? So it seems as if given a sequence of real numbers, it is not always possible to generate a ...
10
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0answers
138 views

Number of graphs with prescribed spectrum

I have a question relevant to the number of graphs with prescribed spectral ratio. Let $A$ be the adjacency matrix of a graph on $n$ vertices. Let $\lambda_i$ be its $i$-th largest (signed) ...
3
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2answers
183 views

Size of MAXCUT from eigenvalues

Is there an interpretation of MAXCUT using eigenvalues of the graph that yields constant factor approximation to MAXCUT? Can the estimates provide sharp lower bound to MAXCUT?
3
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0answers
96 views

Spectral Graph Theory and Matroid Theory

I have just started grad school this year and I have been into Spectral Graph Theory for some time now. Recently I got introduced to Matroid Theory and although I know the field has been around for ...
0
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0answers
130 views

Multi-way spectral partitioning and higher-order Cheeger inequalities

I am reading the paper above by Lee, Gharan and Trevisan, and I am having trouble with lemma 4.8 on page 23. How do we form the set $T_1, \ldots, T_{r'}$ such that $r'>(1-\delta/2)k$ and each set ...
2
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0answers
114 views

Graph has several MST what does it mean combinatorically?

This question is not theoretical, it's about combinatorial meaning. In graph theory there is a notion of complexity of a graph, which is equal to the number of spanning trees in a graph, which ...
1
vote
1answer
188 views

Second eigenvalue and the last eigenvalue

Note : All eigenvalues that I would referring to below would of the adjacency matrix of the graph My question arises from having read about Expander Graphs from a few different sources. The most ...
16
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1answer
572 views

Count the number of spanning trees fast

Let $t(G)$ denote the number of spanning trees in a graph $G$ with $n$ vertices. There is an algorithm that computes $t(G)$ in $O(n^3)$ arithmetic operations. This algorithm is to compute ...
12
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0answers
133 views

Complexity to compute the eigenvalue signs of the adjacency matrix

Let $A$ be the $n\times n$ adjacency matrix of a (non-bipartite) graph. Assume that we are given the amplitudes of its eigenvalues, i.e., $|\lambda_1|=a_1,\ldots, |\lambda_n|=a_n$, and we would like ...
11
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0answers
135 views

Generating a random graph with constraints on spectrum

Consider two sequences $u_1 \geq u_2 \geq ... \geq u_n$ and $l_1 \geq l_2 \geq ... \geq l_n$ with $u_i \geq l_i$ for every $i$. Let $\mathcal{G}(l_{1:n},u_{1:n})$ be all undirected unweighted simple ...
4
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2answers
212 views

Optimal upper bound on the number of non-isomorphic graphs with certain parameter

What are the optimal (or best known) bounds (preferably exact or else asymptotic but not expectation on random graphs) on the number of non-isomorphic (unlabelled) simple (no self-loop), undirected ...
5
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0answers
306 views

Which problems in graph theory can be stated as quadratic programs?

There seem to be many very interesting problems in graph theory that can be written in the form of maximizing/minimizing a quadratic form on either the Adjacency ${\bf A}$ or the Laplacian matrix ...
8
votes
1answer
213 views

Bounds on the smaller eigenvalues of the adjacency matrix of a graph

Are there any known (non-trivial) bounds (combinatorial in nature, based on poly-time computable properties of a graph) on the third, down to the smallest, eigenvalue of an (un-weighted) adjacency ...
8
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0answers
111 views

Extension of Cheeger's inequality with distinguished vertices

The standard Cheeger's inequality for graph $G$ states that $\frac{1}{2}$ $\lambda$ < $\phi(G)$ < $\sqrt{2\lambda}$ where $\lambda$ is the second smallest eigenvalue of the normalized ...
11
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0answers
134 views

expansion with respect to p-norms for p other than 2

Suppose I have an $d$-regular expander graph with $n$ vertices, where the stochastic version of its adjacency matrix $A$ (with entries $1/d$ and zero) has second eigenvalue $\lambda$. Let $x \in ...
9
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0answers
372 views

Sampling from Multivariate Gaussian with Graph Laplacian (inverse) Covariance

We know from e.g. Koutis-Miller-Peng (based on work of Spielman & Teng), that we can very quickly solve linear systems $A x = b$ for matrices $A$ that are the graph Laplacian matrix for some ...
5
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2answers
759 views

In a resistor network, is there any relation between the shortest path and the maximum electric current path?

Consider a shortest path problem between the source $s$ and sink $t$ in an undirected weighted graph. There's a well known algorithm such as Dijkstra's algorithm that solves this problem. Naturally, ...
8
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0answers
278 views

Combinatorial method for computing the largest eigenvector of the adjacency matrix of a graph

Given a connected and non-bipartite graph $G=(V,E)$ with vertex set $V=\{1,\cdots, n\}$, let $A$ denote its adjacency matrix and let $deg(i)$ denote the degree of vertex $i$. Let $D$ be a diagonal ...
0
votes
1answer
171 views

Estimating graphs using random cuts

How easy is it to estimate a graph by observing only a few random cuts? Is there prior work related to this? I did google but could not find anything concrete. Any help would be appreciated. Thanks.
17
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0answers
226 views

Geometric picture behind quantum expanders

(also asked here, no replies) A $(d,\lambda)$-quantum expander is a distribution $\nu$ over the unitary group $\mathcal{U}(d)$ with the property that: a) $|\mathrm{supp} \ \nu| =d$, b) $\Vert ...
8
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2answers
283 views

Applications of Spectral Graph Theory in Information and Coding Theory

I wanted to find out what are some application of SGT in the area of information and coding theory and maybe communications. The most related that comes to mind is the work on Expander Codes Michael ...
5
votes
1answer
301 views

Bipartite maximum matching size from eigenvalues

Supposing we know the adjacency matrix $\mathcal{A}_{G}$ of a given regular (or irregular) bipartite graph $G$. Are there good lower and upper bounds to the size of maximum matching from the graph's ...
9
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0answers
202 views

Spectral gap for random bipartite regular graphs

For a graph $G$, let its Laplacian be $\Delta =I − D^{−1/2}AD^{−1/2}$, where $A$ is the adjacency matrix, $I$ is the identity matrix and $D$ is the diagonal matrix with vertex degrees. I'm ...
2
votes
1answer
643 views

condition to make an adjacency matrix positive semidefinite

I would like to ask how can we transform an adjacency matrix of a graph into a positive semidefinite matrix. Of course, we could set self loops, but I do not know of any result indicating how we can ...
10
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2answers
301 views

Maximum imbalance in a graph?

Let $G$ be a connected graph $G = (V,E)$ with nodes $V = 1 \dots n$ and edges $E$. Let $w_i$ denote the (integer) weight of graph $G$, with $\sum_i w_i = m$ the total weight in the graph. The average ...
7
votes
1answer
444 views

Effect of different graph operations at algebraic connectivity of graph laplacian?

The algebraic connectivity of a graph G is the second-smallest eigenvalue of the Laplacian matrix of G. This eigenvalue is greater than 0 if and only if G is a connected graph. The magnitude of this ...
23
votes
4answers
656 views

Proofs obtained only through spectral graph theory

I have an increasing interest in spectral graph theory, which I find fascinating, and I've started collecting a few documents that I have yet to read more thoroughly than what I so far have. However, ...
7
votes
2answers
276 views

Terminology for sparse cuts in graphs

I have found some ambiguity in how the graph parameters edge-expansion, uniform sparsest cut and conductance are defined and denoted. My questions are: what are the definitions that best match the ...
20
votes
3answers
669 views

Reverse Graph Spectra Problem?

Usually one constructs a graph and then asks questions about the adjacency matrix's (or some close relative like the Laplacian) eigenvalue decomposition (also called the spectra of a graph). But what ...
27
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2answers
543 views

Papers to credit for spectral partitioning of graphs

If $G=(V,E)$ is an undirected $d$-regular graph and $S$ is a subset of the vertices of cardinality $\leq |V|/2$, call the edge expansion of $S$ the quantity $\phi(S) := \frac {Edges(S,V-S)}{d\cdot ...
8
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2answers
360 views

Spectral techniques for genus of a graph

A generic question: are there any spectral techniques to estimate the genus of a graph? I am interested in bipartite graphs.
20
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5answers
957 views

Introduction to spectral graph theory

What are the basic references? Are there any good, high-level surveys of SGT and its applications to CS in general and machine learning more specifically?