For questions on spectral graph theory is the study of properties of a graph in relationship to the characteristic polynomial,...

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2answers
78 views

Upper bound on the difference between two elements of an eigenvector

Let $W$ be the non-negative, symmetric adjacency/affinity matrix for some connected graph. If $W_{ij}$ is large, then vertex $i$ and vertex $j$ have a heavily weighted edge between them. If $W_{ij} = ...
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0answers
56 views

when does a graph with normalized laplacian have a uniform degree distribution?

Consider the graph $G(A)$ with A as its adjacency matrix. Let $L$ be its Laplacian and $L_{sym} = D^{\frac{1}{2}}LD^{\frac{1}{2}}$ be the normalized Laplacian. Now let $A(L_{sym}) = I - L_{sym}$ ...
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1answer
40 views

For which spectrum is the following result for?

I don't know if I am the only one that feels this, but damn Spectral Graph Theory needs some notation change... Maybe it is because I am not so experienced in the field yet, but man every time I read ...
2
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0answers
151 views

Bounds on the maximum eigenvalue of the adjacency matrix of a graph.

I managed to proof the following result for the maximum eigenvalue: $ d_{avg}\leq \lambda_{max} \leq \Delta(G) $ where $d_{avg}$ is the average degree of the graph while $\Delta(G)$ is the maximum ...
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1answer
84 views

What graph Laplacians commute

I know that the graph Laplacian of a fully connected graph commutes with the Laplacian of any other graph. Is there any theorem stating something similar about some more general family of graphs? ...
2
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1answer
105 views

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 ...
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2answers
119 views

Adjacency, Laplacian and Maximum Degree

I am relevantly new to http://math.stackexchange.com/ and this will be my first question, although I have been lurking around for some time now! So, pardon me if I am missing any posting etiquette ...
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2answers
61 views

Graphs with zero spectrum / nilpotent symmetric matrices

Is there a graph theoretic characterization of those graphs with zero spectrum? Alternatively, can one at least characterize all symmetric nilpotent (complex) matrices, so that one could recognize ...
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1answer
88 views

Prove $MM^t=A+kI$ for matrices associated to graphs

How can I prove that $MM^t=A+kI$ for incidence matrix $M$ and adjacency matrix $A$ of a $k$-regular graph with $n$ vertices? It is easy to see that $MM^t$ is an $n\times n$-matrix (like $A$), ...
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1answer
251 views

Adjacency matrix defines a distance metric

Let $A$ be adjacency matrix of a graph (perhaps weighted). Prove that \begin{equation} \sum_i \sum_j A_{ij} (f_i- f_j)^2 = \mathbf{f}^T L \mathbf{f} \end{equation} where $\mathbf{f}$ holds values of ...
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1answer
127 views

Eigenvalues of graph's laplacians

I'm trying to tackle a question from a homework assignment and one of the problems concerns the relation between eigenvalues of a graph's laplacian and its complement's laplacian. The relation is: ...
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2answers
87 views

Spectrum of a 3-regular graph

Let $D_n$ be the following graph on $2n$ vertices: $V=\mathbb{Z}_n\times\{0,1\}$ and $E=\{(i,j)(i+1,j): i\in \mathbb{Z}_n,j\in \{0,1\}\}\cup\{(i,0)(i,1):i\in\mathbb{Z}_n\}$. What is the spectrum of ...
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1answer
363 views

Spectrum of the cycle graph $C_n$

I am trying to find out the spectrum (the collection of eigenvalues) with their multiplicities of the cycle graph $C_n$. Assuming that $X=\pmatrix{x_1\\x_2\\\vdots\\x_n}$ is the eigenvector, by ...
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0answers
44 views

spectral graph theory with “potentials”

Let G be an undirected graph with bounded degree and n vertices. Let L[G] be the corresponding graph Laplacian, which is a symmetric $n \times n$ matrix. Let V be an $n \times n$ diagonal matrix. I am ...
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6answers
2k views

Motivation for spectral graph theory.

Why do we care about eigenvalues of graphs? Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so ...
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1answer
119 views

Eigenvalues of a connected graph $G$ are greater than or equal to $-1$ iff $G$ is perfect?

Consider $P_G$ as the characteristic polynomial of the adjacency matrix of the connected graph $G$. It is easy to prove that $P_{K_n}(x)=(x-n+1)(x+1)^{n-1}$, so all of the eigenvalues of a perfect ...
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3answers
222 views

How to find the spectrum of the hypercube?

I want to find the proof of the spectrum of the hypercube
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0answers
151 views

Is there a simple interpretation of the eigenvectors of a graph (visualizable?)?

I want to understand eigenvectors obtain from graphs (adjacency matrices) in an analogous way as they are interpreted from principal component analysis of a set of images, which is easy:Eigenfaces ...
4
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1answer
253 views

Intuitive interpretation of the adjacency matrix as a linear operator.

Naturally we can describe graphs via tables of "yes there is an edge" or "no there is not" between each pair of vertices, so the definition of an adjacency matrix is easily understood. Thinking of ...
5
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1answer
121 views

Laplacians, Diagonal Perturbations

Setup: Consider a Laplacian (or Kirchoff) matrix $L = L^T \in \mathbb{R}^{n \times n}$ corresponding to a weighted, undirected and connected graph. That is, a matrix with $L_{ij} \leq 0$ for $i\neq j$ ...
2
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1answer
275 views

The number of connected components of a $k$-regular graph equals the multiplicity of k

In similar vein to this question, I am trying to understand the proof of the fact that in a $k$-regular graph, the multiplicity of the eigenvalue $k$ equals the number of connected components. The ...
3
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2answers
1k views

Lowest Eigenvalue of a positive semi-definite matrix

I am reading a paper on spectral graph theory. Let us say we have an adjacency matrix W and a degree matrix D. We construct a Laplacian matrix, L defined as: L = D - W The paper claims that L is ...
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1answer
138 views

Tighter bound on a spectrum of a matrix obtained by partial diagonal shift

A matrix $V\in\mathbb{R}^{n\times n}$ is a full symmetric matrix with negative off-diagonal elements summing (in absolute value) to diagonal elements. Suppose that $V$ has all entries, and let ...
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1answer
200 views

Strongly connected graph associated with a matrix

This type of matrices $L$ is called Leslie type matrices in Population dynamics: $$L = \begin{pmatrix} f_{11} & f_{12} & f_{13} & \dotsm & f_{1,i-1} & f_{1,i}& \dotsm & ...
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3answers
1k views

Eigenvalues of a bipartite graph

Let $X$ be a connected graph with maximum eigenvalue $k$. Assume that $-k$ is also an eigenvalue. I wish to prove that $X$ is bipartite. Now if $\vec{x}=(x_1,\cdots ,x_n)$ is the eigenvector for ...
2
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1answer
288 views

Connectedness of a regular graph and the multiplicity of its eigenvalue

Suppose $X$ is a $k$-regular graph with adjacency matrix $A$. I wish to show that if $k$ has multiplicity $1$ as an eigenvalue of $A$ then $X$ is connected. By way of contradiction I assume that X is ...
4
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0answers
75 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 ...
4
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1answer
162 views

Questions on fractional Laplacian graph spectra

Both the signed ($D-A$) and unsigned ($D+A$) Laplacian are of interest in spectral graph theory, see eg Cvetkovic: Bibliography on the signless Laplacian eigenvalues: first one hundred references. ...
4
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1answer
216 views

characteristic polynomial of the adjacency matrix of a tree

I have read that if $A$ is the adjacency matrix of a tree $T$, then we have that $$\det(\lambda I - A) = \sum_{k=0}^{\lfloor n/2 \rfloor} (-1)^k N_k(T) \lambda^{n-2k} $$ where $N_k(T)$ is the number ...
2
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1answer
3k views

Plotting a one-sided amplitude spectrum

I have a continuous signal $x(t)$ such that $$x(t)=12\cos(6\pi t)+6\cos(24\pi t)+3\cos(30 \pi t)$$ and is asked to sketch a $1$-sided Amplitude Spectrum of the signal $x(t)$ if sampled above the ...
3
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2answers
322 views

A finite graph G is $d$-regular if, and only if, its adjacency matrix has the eigenvalue $λ = d$

Show that a graph $G$ finite with $n$ vertices is $d$-regular if, and only if, the vector with all the coordinates equals to 1 is eigenvetor from eigenvalue $λ = d$ from the adjacency matrix $A$ ...
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1answer
625 views

Eigenstructure of discrete Laplacian on uniform grid

The discrete Laplacian of a graph is the matrix $L = D - A$ where $D$ is a diagonal matrix with $d_{ii}$ being the degree of $v_i$, and $A$ is the usual adjacency matrix. Is there anything known ...
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0answers
47 views

Properties of a generalized graph

I'll start with formulating my problem and then ask my question: To generalize a graph $Ga = (Va,Ea)$, we partition its nodes into disjoint sets. The elements of a partitioning $V$ are subsets of ...
4
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1answer
229 views

Spectral graph theory and connected components of graphs

We know that multiplicity of least eigenvalue of laplacian matrix of graph gives us number of connected components in graph.What is intuition behind this theorem? How do we know that this works in ...
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3answers
1k views

Significance of eigenvalue

When I represent a graph with a matrix and calculate its eigenvalues what does it signify? I mean, what will spectral analysis of a graph tell me?
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1answer
3k views

What do the eigenvectors of an adjacency matrix tell us?

The principal eigenvector of the adjacency matrix of a graph gives us some notion of vertex centrality. What do the second, third, etc. eigenvectors tell us? Motivation: A standard information ...
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5answers
2k views

Spectrum of adjacency matrix of complete graph

Fooling around in matlab, I did an eigenvalue decomposition of the adjacency matrix of $K_5$. ...
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0answers
127 views

Spielman. Spectral Graph Theory Proposition

Spielman says in Lecture 3: Laplacians and Adjacency Matrices Fiedler’s Theorem will follow from an analysis of the eigenvalues of tri-diagonal matrices with zero row-sums. These may be viewed as ...
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1answer
175 views

Relationship Between Eigenvalue and Degree

I have a question regarding Rayleigh quotient. It's well known that maximal eigenvalue can be found by $$\lambda_{\max}(M)=\max_{x\neq 0}\frac{x^{T}Mx}{x^{T}x}.$$ Using this how to prove that ...
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0answers
71 views

algebraic connectivity of the giant component [closed]

In percolation theory there is this idea of a giant component, and I am curious what is known about its algebraic connectivity. I looked on google but I was not able to find anything particularly ...
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1answer
372 views

Spielman's proof of graph connectivity

I use Spielman's lectures on course Spectral Graph Theory I have few question regarding Lecture 2. The Laplacian, especially Lemma 2.3.1 (Graph connectivity). Please, help me to make it a little bit ...
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1answer
215 views

Proof of Algebraic connectivity

I am very curious about the proof of Algebraic connectivity Algebraic connectivity: The algebraic connectivity of a graph $G$ is the second-smallest eigenvalue of the Laplacian matrix of $G$. ...
8
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1answer
178 views

Two formulas for the minimal eigenvalue of a graph

Hello again everybody, I'm reading Fan Chung's monograph Spectral Graph Theory. In it, she has two formulas for the minimal eigenvalue of a graph. She doesn't explain why they're equivalent, and I'm ...
12
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1answer
456 views

What does the minimal eigenvalue of a graph say about the graph's connectivity?

I'm reading Fan Chung's Spectral Graph Theory, and I'm now in chapter 2. There, Chung proves Cheeger's inequality, which is that $2h_G \geq \lambda_1 > h_G^2/2$ for any graph $G$. To me, this ...
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0answers
455 views

Is there any relation between the principal eigenvalue of sub matrix and the original matrix?

I am wondering whether there is any relation between principal eigenvalue of sub matrix and the original matrix. In fact I am facing a problem which is to select $n$ rows and $n$ columns from the ...
3
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1answer
360 views

Does an $n\times n$ adjacency matrix of a scale-free network graph have $n$ distinct eigenvalues?

Question updated Suppose that I have an $n\times n$ adjacency matrix $\mathbf{A}$ of a simple graph $G$ where the entry $(i,j)$ represent the number of edges between node $i$ and $j$ in $G$. Note ...
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1answer
210 views

Justifying a pair of inequalities involving the exponential function

I'm reading Fan Chung's Spectral Graph Theory. There's a pair of inequalities I don't know how to justify. Chung doesn't attempt to explain them, so maybe they're very obvious. Example 1.19 on page ...
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2answers
169 views

Question from section 1.5 of Chung's Spectral Graph Theory

I'm (slowly) reading Fan Chung's Spectral Graph Theory. At the moment, I'm in section 1.5 which is about eigenvalues and random walks. There's a small technical point that puzzles me. The context ...
2
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1answer
171 views

Kinks in the eigenvalue spectrum of short range lattices

Take a periodic one-dimensional lattice of size $N$ with $2k$ nearest neighborers. That is, vertex $i$ is connected to $i+1,i+2,...,i+k$ and $i-1,i-2,...i-k$ (with the understanding that the indices ...