All Questions
Tagged with sp.spectral-theory spectral-graph-theory
46
questions
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51
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What do you call this class of matrices with a unique positive eigenvalue associated to a graph?
I am looking for the name of a class of symmetric matrices $M\in\Bbb R^{n\times n}$ that I can associate to a (finite simple) graph $G=(V,E)$ with $V=\{1,...,n\}$ and that have the following ...
- 11.9k
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0
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76
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Graph energy and spectral radius
Suppose $G$ is a simple graph of order $n$ with eigenvalues $\lambda_1\geq \cdots\geq \lambda_n$. I've encountered the quantity $L=\big\vert |\lambda_1|-|\lambda_2|-\cdots-|\lambda_n|\big\vert$. Note ...
- 171
2
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1
answer
111
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Non-regular cospectral graphs with same degree sequences
I am looking for a large family (infinite pairs) of cospectral graphs with these condtions:
The graphs are non-regular,
Minimum degree is greater than $1$,
The degree sequences of these cospectral ...
- 4,738
3
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0
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250
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Eigenvalue bounds and triple (and quadruple, etc.) products
Very basic and somewhat open-ended question:
Let $A$ be a symmetric operator on functions $f:X\to \mathbb{R}$, where $X$ is a finite
set. Assume that the $L^2\to L^2$ norm of $A$ is $\leq \epsilon$, i....
- 19.1k
60
votes
5
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10k
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Intuitively, what does a graph Laplacian represent?
Recently I saw an MO post Algebraic graph invariant $\mu(G)$ which links Four-Color-Theorem with Schrödinger operators: further topological characterizations of graphs? that got me interested. ...
- 250
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3
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973
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Algebraic graph invariant $\mu(G)$ which links Four-Color-Theorem with Schrödinger operators: further topological characterizations of graphs?
30 years ago, Yves Colin de Verdière introduced the algebraic graph invariant $\mu(G)$ for any undirected graph $G$, see [1]. It was motivated by the study of the second eigenvalue of certain ...
- 6,757
5
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2
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1k
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Fiedler vector, what else?
In the spectral analysis of a graph with 1 connected component, the first non-trivial eigenvector (corresponding to the non-zero smallest eigenvalue) is also called the Fiedler vector. This vector is ...
2
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0
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74
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Upper bound for smallest eigenvalue of infinite family of graphs
Let $\left\{G_{n}\right\}_{n=1}^{\infty}$ be a sequence of regular simple connected graphs with at least one edge such that $G_i$ is an induced sub-graph of $G_{i+1}$ and is not equal to $G_{i+1}$.
...
- 133
2
votes
1
answer
158
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Anderson localization for Bernoulli potentials on half-line
Anderson localisation for (discrete) Schrödinger operators with Bernoulli potentials on $l^2(\mathbb{Z})$ was proven in
https://link.springer.com/article/10.1007/BF01210702
I am wondering if there ...
- 223
2
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0
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65
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Infinite trees whose spectrum has more than 3 connected components
I was wondering whether there exists any infinite tree $T$ such that the action of $\mathit{Aut}(T)$ on the set of vertices $V=V(T)$ has finitely many orbits, and whose spectrum $\sigma(T)$ has ...
1
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1
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111
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Spectral bound for maximum clique $k(G)$ in a permutation graph
Let $\pi \in S_n$ be an arbitrary permutation. By permutation graph, we refer to a simple graph with nodes $[n]$ and edges that connect pairs of nodes that appear sorted in $\pi$. Formally, $G=(V=[n],...
- 175
4
votes
2
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1k
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What's the full assumption for Laplacian matrix $L=BB^T=\Delta-A$?
Graph with no-selfloop, no-multi-edges, unweighted.
directed
For directed graph Adjacency matrix is a non-symmetric matrix $A_{in}$ considering indegree or $A_{out}$ considering outdegree. Degree ...
- 211
3
votes
1
answer
386
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Graph Fourier transform definition
I have a question about the definition of the graph Fourier transform. Let me start with definition.
Let $A$ be the adjacency matrix of a graph $G$ with vertex set $V = \{1, 2, \dots, n\}$. The ...
- 41
1
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1
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209
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Graph Laplacian Operator
Consider the linear operator $\mathbb{L} : L^2([0,1])\to L^2([0,1])$ defined by
$$
(\mathbb{L}f)(x) = \int_0^1 xy(f(x)-f(y)) \mathrm{d}y
$$
for all $f\in L^2([0,1])$ and $x \in [0,1]$. Is $\mathbb{L}$...
- 31
3
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1
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233
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Local-Global Principle in Graph Spectrum
The question is a bit vague, but any ideas/directions will be appreciated.
Let us fix an $n$-vertex $d$-regular graph $G=(V,E)$. As I understand it, the eigenvalues of the adjacency matrix $A$ of $G$ ...
- 939
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0
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232
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Does the zeta regularized Laplacian determinant measure the volume of some parameter space? How many "spanning trees" on a manifold?
Let $(M,g)$ be a Riemannian manifold, with Laplacian $\Delta$. If $\lambda_i$ are the nonzero eigenvalues of $\Delta$, we can define the zeta function $\zeta(s) = \Sigma \lambda_i^{-s}$. By analytic ...
- 1,422
5
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1
answer
415
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What is $e^{- \zeta_{\Delta} '(0)}$ for a $\Delta$ the Laplacian of a manifold?
For a connected, finite graph $G$, let $\lambda_1, \ldots, \lambda_n$ denote the nonzero eigenvalues of the graph Laplacian. We define $\zeta_G = \Sigma_{i = 1}^n \lambda_i^s$.
Then Kirkoffs Matrix-...
- 1,422
4
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1
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369
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Distribution of eigenvectors and eigenvalues for random, symmetric matrix
Consider two simple, undirected graphs with adjacency matrices ${\bf A}$ and ${\bf A'}$. Let ${\bf P} = {\bf A'} - {\bf A}$. Thus, ${\bf P}$ is symmetric and always 0 along the diagonal.
Let $f({\bf ...
- 43
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0
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48
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Relation between nullity of components to its parent graph
Let $G$ be an undirected graph and the corresponding adjacency matrix be $A$. Let $v$ be a cut-vertex of $G$. Let $G_1, G_2,\dots, G_k$ are the connected components of the induced graph $G-v$ ( the ...
- 620
6
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2
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229
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Generalised Isospectrality of Graphs
Q: Is there a graph matrix-representation (not necessarily an $n \times n$ matrix for an $n$-graph) such that isospectrality implies graph-isomorphism? For instance, would the simple distance-matrix ...
- 303
5
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1
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863
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The spectrum of the discrete Laplacian
Consider a connected (we define connected components by defining the set of vertices where every vertex has one neighbour) sublattice $V$ of the square lattice $V \subset\mathbb{Z}^2.$
On this we ...
- 53
2
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3
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588
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What are the eigenvectors of the graph Laplacian of a Johnson graph J(n,k)?
The spectral properties of Johnson graphs are known. However I also like to know more about their eigenvectors. Are there any results on the eigenvectors of the graph Laplacians of these Johnson ...
- 131
5
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1
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972
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Intuition on Kronecker Product of a Transition Matrix
Let $T$ be a $N\times N$ transition matrix for a markov chain with $N$ states. Thus $T_{ij}$ is the probability of transition from state $i$ to state $j$ (and thus rows summing to one). Now consider ...
- 1,371
5
votes
2
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949
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Lower bound for smallest eigenvalue of symmetric doubly-stochastic Metropolis-Hasting transition matrix
For my master's thesis research, I stumbled upon a question concerning the Metropolis-Hasting transition matrix $W$.
Context $\quad$
Let me start with some context. I consider connected undirected ...
- 51
14
votes
1
answer
2k
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Reasons for difficulty of Graph Isomorphism and why Johnson graphs are important?
In http://jeremykun.com/2015/11/12/a-quasipolynomial-time-algorithm-for-graph-isomorphism-the-details/ it is mentioned:
'In discussing Johnson graphs, Babai said they were a source of “unspeakable ...
user76479
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226
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Lp norm estimates for the inverse of the Laplacian on a graph
I am looking at a finite connected graph and I would like to know what is the best [i.e. largest] constant $\lambda_p$ in
$$
\sum_x f(x) =0 \implies \| \Delta^{-1} f\|_{\ell^p} \leq \lambda_p^{-1} \|...
- 4,342
4
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124
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An inequality from the "Interlacing-1" paper
This question is in reference to this paper, http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf (or its arxiv version, http://arxiv.org/abs/1304.4132)
For the argument to ...
3
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0
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41
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About the partial expectation polynomials in the Interlacing-I paper and perfect matchings
I am thinking of the polynomials $f_{s_1,s_2,..,s_k}$ as in the definition 4.3 in this paper http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf
In the use of these ...
0
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0
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160
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$l_{\infty}$ norms of matrix perturbations
Say $B$ is a real symmetric matrix of dimension $n$ and $A$ is another real symmetric matrix of the same dimension.
What needs to be the bounds on (which?) norm of $B$ to ensure that $\lambda_{max}(...
- 1,863
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0
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154
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Interpreting (Fiedler) spectral bisectioning
I would appreciate help on how to interpret the results of spectral bisectioning of a graph.
Given a $G=(V,E)$ with size $N$ represented by $Q$ its Laplacian matrix where the eigenvalues are ordered ...
- 355
2
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1
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1k
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About distinct eigenvalues of a graph
if a graph with adjacency matrix $A$ and Laplacian $L$ has $k$ distinct eigenvalues then does this fact naturally help define or prove existence of a polynomial $p$ of degree $k-1$ such that $[p(A)]_{...
- 1,863
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2
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590
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 ...
- 1,863
1
vote
2
answers
280
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Graph lifts and representation theory
Is there any connection known between the two?
One can naturally define lifts of graphs by groups like $\mathbb{Z}_k$ and hence I wonder if representation theoretic properties can be used to say ...
- 1,863
0
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1
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218
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When is a $2$-lift of a graph connected? [closed]
Let $\ (V\ E)\ $ be a graph, i.e. $\ E\subseteq\binom V2.\ $ A $2$-lift pattern of a graph is a function $\ e:E\rightarrow\{-1\,\ 1\}.\ $ The induced 2-lift is defined as the graph $\ V\times\{-1\,\ ...
- 1,863
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1
answer
211
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Laplacian spectrum of $2-$lifts of graphs
We know that a $2-$ lift of a graph is specified by a $\pm 1$ assignment on the edges of the graph ( given as a signing matrix) denoting which edge is to be duplicated by the identity permutation on ...
- 1,863
4
votes
0
answers
580
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 ...
- 1,863
13
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3
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4k
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What is a "Ramanujan Graph"?
I noticed an apparent conflict in the definition in literature about what is a "Ramanujan graph, which I was wondering if someone could kindly clarify.
(1)
The Hoory-Linial-Wigderson review on ...
- 1,863
6
votes
2
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533
views
Do perfect matching(s) have signatures in the graph eigenvalues?
If the edges of a bipartite graph are such that they can be seen as a disjoint union of perfect matchings then will this somehow reflect in the eigenvalues of the Laplacian?
It would be helpful to ...
- 1,863
11
votes
3
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830
views
Are these three different notions of a graph Laplacian?
I seem to see three different things that are being called the Laplacian of a graph,
One is the matrix $L_1 = D - A$ where $D$ is a diagonal matrix consisting of degrees of all the vertices and $A$ ...
- 1,863
10
votes
3
answers
653
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Is there a continuous analogue of Ramanujan graphs?
I think it might help to think of the following definition of a Ramanujan graph - a graph whose non-trivial eigenvalues are such that their magnitude is bounded above by the spectral radius of its ...
- 545
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votes
1
answer
874
views
Can the graph Laplacian be well approximated by a Laplace-Beltrami operator?
It seems rather well known that given a Laplace-Beltrami operator $\mathcal{L}_{M}$ on a manifold $M$ we can approximate its spectrum by that of a graph Laplacian $L_{G}$ for some $G$ (where $G$ is ...
- 6,832
1
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0
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130
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interpretation of generalized eigenvalue/vectors in spectral graph theory [closed]
Let us say I have a symmetric graph adjacency matrix A, a degree matrix D, a laplacian L (D-A). I have a generalized eigenvalue equation $Av=\lambda Lv$. Does the eigenvalue/vectors produced in this ...
- 131
2
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0
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168
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Recovering a partition from spectral properties of the graph Laplacian
Let $G$ be a weighted graph with vertices $V$. Let $W$ be its real-valued, non-negative, $|V|\times|V|$ adjacency/affinity matrix. Let $L = \mathrm{diag}(W\mathbf1)-W$ be the (unnormalized) graph ...
1
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1
answer
388
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Spectrum of composition of graphs( lexicographic product)
I was wondering how to relate the spectra of the composition of two graphs in term of the factors...someone can help me?
- 119
0
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0
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142
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Global solution for spectral clustering
I used spectral clustering for directed graphs suggested by Dengyong Zhou paper to partition the graph.I selected the eigen vectors corresponding to k largest eigen values and then I use kmeans or FCM ...
- 13
2
votes
1
answer
2k
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Eigenvalues of directed Laplacian matrix $L$ and $DL$, where $D$ is a diagonal matrix with positive entries
I have a weighted Laplacian matrix $L$ of a strongly connected directed graph and a diagonal matrix $D$ with positive entries. Since the graph is directed, $L$ is non-symmetric real. Further, since ...
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