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1answer
54 views

Finding the smallest max eigenvalues for related matrices?

While messing around with a spectral approach to a graph coloring question, I happened upon a type of problem I hadn't seen before. Suppose you have two symmetric $n$ x $n$ matrices in the form ...
0
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1answer
22 views

Top and bottom power spectral density of a height profile

Imagine I have a simple 1D height profile which is NOT symmetric. Now, what is truly important for me is to know what are the frequency content of the top profile (i.e. a cut profile above the ...
0
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1answer
22 views

If $G$ is connected then $\lambda_2 < \lambda_1$.

Let $G=(V,E)$ be an $n$-vertex , undirected graph with maximum degree $d$, then how to prove the following result. If $G$ is connected then $\lambda_2 < \lambda_1$. where $\lambda_1 \geq \lambda_2 ...
2
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0answers
110 views

a closed formula to enumerate the self avoiding walks of a graph

Let $G$ be a directed graph with $N$ nodes and weighted adjacency matrix $W $ defined by $$ W_{ij} = \left\{ \begin{array}{cl} w_{ij} & \text{ if } \ i \ \text{ is connected to } j \\ 0 & ...
0
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1answer
27 views

Control principal eigenvector of a row stochastic matrix

I am just trying to consider the classical discrete-time Markov Chain problem. Consider the transition matrix P, which transforms state vector $x(k)$ to $x(k+1)$, satisfying: $x(k+1)$ = $P*x(k)$ It ...
1
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1answer
67 views

Graph Isomorphism algorithm that doesn't always work

I just thought of the following incomplete algorithm for deciding whether two graphs are isomorphic: Let $A$ and $A'$ be adjacency matrices of two graphs. Then for some unitary $U,U’$ and diagonals ...
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0answers
15 views

What is the importance of Laplacian Matrix in spectral clustering

In spectral clustering algorithms, we often use Laplacian matrix of adjacency matrix instead of Adjacency matrix itself. What is the advantage of using Laplacian matrix over simple adjacency matrix?
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0answers
32 views

Spectrum of an infinite graph independent of labelling

Does there exist an infinite graph whose spectrum does not depend upon the labelling of the graph? While evaluating the spectrum, I am considering adjacency matrix of the infinite graph as a bounded ...
2
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1answer
74 views

Eigenvalues of the distance-k graph of a graph

Let $G$ be a (finite, simple, connected) graph. Define the distance-$k$ graph $G_k$ to be the graph with the same vertex set and $x\sim y$ iff $d(x,y)=k$. A graph is integral if all of the eigenvalues ...
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1answer
17 views

Motivations for Shi-Malik Algorithm

So I've been trying to make sense of the clustering algorithm on page 6 of this paper. Are the "first" k eigenvalues they refer to the smallest eigenvalues? What are the $y_i$ exactly? I don't see ...
2
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1answer
36 views

Functions on adjacency matrices

From Norman Biggs, Algebraic Graph theory 2j, p13: The adjacency matrix has a spectral decomposition $A = \sum \lambda_aE_a$, where the matrices $E_a$ are idempotent and mutually orthogonal. (...) It ...
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0answers
113 views

Applications of Cayley Graphs in Physics

I have been recently reading about Cayley graphs and character theory. It is evident that Cayley graphs are very useful tool in theoretical computer science. In physics, Cayley graphs seem do appear ...
3
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1answer
51 views

Show the relationship between the trace and the number of 4-cycles

Let $G$ be a k-regular graph. Show the exact relationship between $tr(A^4)$ and the number of 4-cycles in $G$. I understand how $tr(A^4)$ tells us the total number of closed paths of length 4 in ...
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2answers
73 views

Power series convergence of random walk transition matrix

I would like to find out if $$ \sum_{t=0}^\infty P^t = \left( I- P \right)^{-1} ~,$$ where $P = D^{-1}W ~ $ is a random walk transition matrix. $W$ is a matrix describing weights in a graph and ...
1
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2answers
76 views

Graph Theory - Proof

I am need help to Prove the following statement: Let G be a $k$-regular graph with $n$ vertices and $k \geq 1$. Prove that $G$ does not have an independent set of size greater than $\dfrac{n}{2}$. ...
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0answers
26 views

The lower bound of Cheeger Inequality as the degree goes to infinity

Consider an undirected graph $G(V,E)$ with adjacency matrix $A$ and define the graph Laplacian as \begin{equation} L = D - A \end{equation} where $D$ is a diagonal matrix such that $D(i,i) = d_i$. ...
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0answers
91 views

Does any vertex transitive graph have a bounded eigenvector?

Following up on the negative answer to this question, I would be interested in knowing the answer to the following question, which I cannot seem to find an obvious contradiction to when testing for ...
0
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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: ...
2
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1answer
65 views

Algebraic Combinatorics

Let $K_{r,s}$ denote the complete bipartite graph, defined on $r + s$ vertices $\{v_1,v_2,...,v_r,w_1,...,w_s\}$, with an edge between $v_i$ and $w_j$ for $1 ≤ i ≤ r$ and $1 ≤ j ≤ s$. By ...
1
vote
1answer
68 views

rate of convergence of absorbing markov chain

Let $G$ be a biconnected and non-bipartite graph. I can simulate a random walk on this graph with a markov chain. The stochastic matrix is $M = AD^{-1}$, where $A$ is the adjacency matrix of $G$ and ...
3
votes
2answers
79 views

Is the eigenvectors of vertex transitive graphs bounded

For a connected and regular graph $G$ with degree $ d $ at each vertex and adjacency matrix $A$, the normalized Laplacian of $G$ is defined as $L = I-\frac{1}{d}M$. Let $\psi$ be an eigenvector of $L$ ...
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0answers
76 views

Number of Nodes within a Given Distance from a Node

Suppose we are given a $d$-regular graph $G=(V,E)$ of order $n$. Let $\lambda_2$ be the second-largest eigenvalue of $G$'s adjacency matrix. Does this information help obtaining a lowerbound or ...
2
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0answers
52 views

Interesting Questions in Spectral Graph Theory

In the past, I have worked on few problems in Spectral graph theory and their applications to Physics. I have read parts of Fan Chung's book and Daniel Spielman lecture notes. I really enjoyed the ...
2
votes
1answer
46 views

the solution of matrix polynomials

In order to get the eigenvalues of \begin{equation} P=\left[ \begin{array}{cc} 0_{n\times n} & I_{n\times n} \\ -A & -B% \end{array} \right], \end{equation} where $A$ and $B$ are both $n\times ...
0
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0answers
29 views

Eigen vectors of graph laplacians

I have been reading about spectral graph theory from Daniel A. Spielman's notes. Fiedler’s Nodal Domain Theorem from this note says that : Let $G = (V, E, w)$ be a weighted connected graph, and let ...
2
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0answers
48 views

interpretation of generalized eigenvalue/vectors in spectral graph theory

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 ...
0
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1answer
41 views

Lower bound on the minimum eigenvalue of sum of two matrices

Assume that $A$ is a symmetric positive definite matrix and $B$ is a symmetric (can potentially negative entries). Is the following bound correct? $$\lambda_{min}(A+B)\geq ...
5
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0answers
79 views

Spectral gap of mixture of Markov chains

Context Let $P$ be the transition matrix of an irreducible, aperiodic, discrete-time Markov chain. The spectral gap is given by $$\xi = 1 - \lambda_\max$$ where $\lambda_\max = \max\{\lambda_2, ...
5
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0answers
144 views

Generating a stochastic matrix with a given second dominant eigenvalue

I need a procedure (iterative or otherwise) that, given a positive integer $N$ and a (possibly complex) number $\lambda$ such that $0 < \vert \lambda \vert < 1$, will be able to generate an $N ...
0
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0answers
13 views

Graph partitioning with constraints

Consider having $n$ data points ${{x}_{1}},..,{{x}_{N}}\in {{R}^{D}}$. Given an affinity matrix of the data, $W=[{{w}_{ij}}]$ where ${{w}_{ij}}$ is the affinity measure for data points ${{x}_{i}}$ and ...
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0answers
20 views

Graph Laplacian - Spectral Clustering with Regularization

Assume a graph $G=(V,E)$ where the vertices $V$ are points in ${{R}^{D}}$ with $\left| V \right|=n$. The edges $E$ are represented by a $n\times n$ affinity matrix $W$. Consider the graph Laplacian ...
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0answers
44 views

How can I prove that a particular family of graphs is integral?

I'm working with an infinite family of graphs that seem to always have all integral eigenvalues, and I'd like to find some way to prove that (if it's true). Call the graphs $G_{n,k}$ and define them ...
2
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0answers
49 views

Possible Eigenvalues of Graph

How would one prove (or disprove) that there is no such graph $G$ with $\lambda$ as an eigenvalue? I tried setting up a system of equations to see if it's possible for $-\frac{1}{2}$ to be an ...
0
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0answers
30 views

(Girko's) circular law beyond the standard normal distribution

I found this circular law , which states a uniform distribution of eigenvalues in the unit circle of the complex plane when the eigenvalues are of a asymmetric random matrix of which the elements are ...
1
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0answers
41 views

pierre simon laplace and his knowledge of the (Laplacian) matrices

so as we all know, there is a graph matrix called the Laplacian that is used in some eigenvalue/eigenvector/graph theory/spectral theory problems. i'm wondering if the name of this matrix is ...
1
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0answers
27 views

How to show that the spectral radius of a binary tree approaches exp(1) as the N tends to infinity?

How can I prove mathematically that the spectral radius of a binary tree approaches e as the number of nodes tends to infinity? I mean it is true that the increase in nodes number is exponential but ...
1
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0answers
49 views

Characterization of all matrices with unit spectral radius under constraint

Let $A \in \mathbb{R}^{n \times n}_{\geq 0}$ be a symmetric matrix with positive row sums $\mathbf{d} := A\mathbf{1} > 0$. I am interested in characterizing all those positive diagonal matrices $Z ...
3
votes
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 ...
4
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0answers
159 views

spectral radius (adjacency matrix) of a partitioned graph

Is the following bound on the largest eigenvalue of a partitioned graph known? It seems like it should be but I am not able to find a reference to it. Given a graph $G=(V,E)$ whose vertices are ...
0
votes
1answer
82 views

Compute eigenvalues of a regular graph.

I have a $(q^2+q)(q+1)$-regular graph. Is there some general method to compute the eigenvalues of the adjacency matrix of a $k$-regular graph explicitly? Or could we estimate its eigenvalues? Thank ...
0
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0answers
56 views

Controlling the spectral radius of a matrix product

Given a binary $n \times m$ matrix $B$, I want to find a real $m \times m$ diagonal matrix $D$ that minimizes the spectral radius of $BDB^T$, subject to the constraints: $Tr(D) = 0$. $\|D\|_{F1} = ...
0
votes
1answer
92 views

how do I compute the eigenvectors for spectral clustering from a singular value decomposition?

I am implementing spectral clustering following A tutorial on spectral clustering. After preparing the Laplacian matrix $L^{n \times n}$, I compute the Singular Value Decomposition $U \Sigma V^{*}$. ...
0
votes
2answers
407 views

Rank of adjacency matrix vs rank of graph Laplacian

What is the relation between rank of the adjacency matrix of a graph and rank of the corresponding graph Laplacian matrix?
0
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1answer
57 views

Are these equivalent representations (labelled graph and adjacency matrix)?

This is an example from Wikipedia's page on adjacency matrices, which from the site's format seems to be suggesting equivalence between the simple diagram below, left, and the abstractly represented ...
3
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0answers
84 views

Constructing a directed graph from its spectrum

This is related to the following question from cs theory stack exchange: http://cstheory.stackexchange.com/questions/3742/reverse-graph-spectra-problem So it seems as if given a sequence of real ...
3
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0answers
41 views

Is the upper Cheeger Inequality tight?

The (upper) Cheeger Inequality says: Let $G$ be an unweighted, undirected, regular graph of degree $d$. Let $\lambda_2$ be the second eigenvalue of the Laplacian matrix of $G$, and let $\phi(G)$ ...
27
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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 ...
3
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0answers
131 views

What can we say about two graphs if they have similar adjacency matrices?

Suppose we have two (finite, simple, undirected) graphs, what can we say about these graphs if they have similar adjacency matrices? Observations to begin with: If $G_1$ and $G_2$ are isomorphic, ...
0
votes
0answers
126 views

Variance of the first return time of a simple random walk on an hypercube graph

I am trying to solve this problem.... I have a simple random walk on a $d$-cube (finite graph). At each vertex of the graph, the particle chooses one of $d$ edges equally likely. I need to calculate ...
1
vote
1answer
73 views

cluster validation and determining number of clusters

I want to find number of cluster in the real world data set. So, I validate the spectral clustering by using some indexes as shown in figures below? But as you seen in figures the results are very ...