For questions related to the study of properties of a graph in relationship to the spectral properties of some associated matrix.

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32
votes
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 ...
27
votes
1answer
4k 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 ...
14
votes
1answer
524 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 ...
11
votes
5answers
2k views

Spectrum of adjacency matrix of complete graph

Fooling around in matlab, I did an eigenvalue decomposition of the adjacency matrix of $K_5$. ...
10
votes
1answer
827 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 ...
8
votes
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?
8
votes
1answer
202 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 ...
8
votes
1answer
161 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 ...
7
votes
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 ...
7
votes
1answer
491 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 ...
7
votes
0answers
88 views

What is the intuition behind / How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices ?

Given a set of points $x_1,x_2,...,x_m$ in the euclidean space $\mathbb{R}^n$, we can form a $m$ x $m$ Euclidean Distance Matrix $D$ where $D_{ij}={||x_i-x_j||}^2$. We know a little bit about these ...
6
votes
1answer
104 views

Graph with largest eigenvalue “almost” $\pi$

While doodling recently I found that the largest eigenvalue of the adjacency matrix of the following undirected graph (ignore directions on edges in picture) is "almost" $\pi$. According to octave ...
5
votes
2answers
454 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 $A$ ...
5
votes
1answer
275 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 ...
5
votes
1answer
71 views

Understanding and interpreting graph spectra

I'm not a mathematician, but a geographer trying to get a grasp on some network analysis I'm experimenting with. I have a few questions related to spectral graph theory that a mathematician could help ...
5
votes
1answer
131 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 $i\neq j$ ...
5
votes
1answer
161 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
votes
0answers
306 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 ...
4
votes
1answer
394 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 ...
4
votes
1answer
46 views

Laplacian solvers for inversion of large matrices?

I have a large matrix L of size 400,000 $\times $ 400,000 . I'm using this L matrix in the following way. Lin = L$^{-1}$ C = D - B * Lin * B'; B,D are of appropriate sizes. L matrix is ...
4
votes
1answer
69 views

Is this graph and its spectrum understood?

Consider the graph whose vertices are labelled by the binary representation of the integers from $0$ to $2^{d}-1$ for some $d \in \mathbb{N}$. So its a graph with $2^d$ vertices. An edge exists ...
4
votes
2answers
77 views

Is there anything special about a graph with the golden ratio in its spectrum?

Given a simple connected graph $g$ with adjacency matrix $\mathbf{A}$. Let the spectrum $\lambda_1 < \lambda_2 < \ldots < \lambda_N$ be the eigenvalues of the equation $\mathbf{A} v=\lambda ...
4
votes
1answer
272 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 ...
4
votes
1answer
117 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? ...
4
votes
1answer
219 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
votes
0answers
51 views

What do the eigenvalues/vectors of a metric describe?

Given a finite metric space $(X = \{ x_i \}_{i=1}^n,d)$, one can form the matrix $A$ of pairwise distances $a_{ij} = d(x_i, x_j)$. What does the eigenspectrum of this matrix say about the metric $d$? ...
4
votes
0answers
216 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 ...
4
votes
0answers
122 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 ...
4
votes
0answers
79 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 ...
3
votes
2answers
70 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 ...
3
votes
1answer
64 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 ...
3
votes
1answer
474 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 ...
3
votes
2answers
189 views

How adjacency matrix shows that the graph have no cycles?

Let $G$ a directed graph and $A$ the corresponding adjacency matrix. Let denote the identity matrix with $I$. I've read in a wikipedia article, that the following statement is true. Question. Is it ...
3
votes
2answers
128 views

Are 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 $L$ ...
3
votes
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 ...
3
votes
3answers
105 views

Why Laplacian Matrix need normalization and how come the sqrt of Degree Matrix?

I am new here. If I do any rough, please forgive me. My question: Why Laplacian Matrix need normalization and how come the sqrt-power of Degree Matrix? The ...
3
votes
0answers
82 views

Clustering with SVD

I'm trying to do some clustering on a graph, which is represented by an adjacency matrix $B = A^2$, where $A$ is symmetric. I tried several methods like taking the eigenvectors of the Laplacian $L = ...
3
votes
0answers
110 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 ...
3
votes
0answers
58 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 $\phi(G)$ ...
3
votes
0answers
161 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, ...
2
votes
3answers
203 views

Can I find the connected components of a graph using matrix operations on the graph's adjacency matrix?

If I have an adjacency matrix for a graph, can I do a series of matrix operations on the adjacency matrix to find the connected components of the graph?
2
votes
1answer
387 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 ...
2
votes
2answers
54 views

How to use spectral graph theory to get a measure for graph symmetry?

I looked at graphs, like $K_{12}$ or Frucht's graph and wondered if their spectrum, more specific the degenercies of their eigenvalues, is a mesaure for the (a)symmetry of the corresponding graph? ...
2
votes
1answer
71 views

Is there any graph property which is equivalent to that the spectral radius of its adjacency matrix is less then $1$?

Let $G$ be a directed graph and $A$ the corresponding adjacency matrix. I'll denote with $\rho$ the spectral radius, and with $I$ the identity matrix. What can we say about $G$ when the spectral ...
2
votes
1answer
423 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 ...
2
votes
1answer
105 views

Connection between the Tutte and characteristic polynomials?

Both the Tutte polynomial $T_G(x,y)$ and the characteristic polynomial $\phi_G(x)$ encode a great amount of structure of the input graph $G$. I've read somewhere that the Tutte polynomial has a kind ...
2
votes
1answer
106 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 ...
2
votes
1answer
140 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 ...
2
votes
1answer
184 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 ...
2
votes
1answer
58 views

About Cayley graphs on finite fields.

If one is given $n$ vectors of length $n$ $\in \mathbb{F}_{p^k}^n$ for some prime number $p$ and $k \in \mathbb{Z}^+$ then how can one check if they are linearly independent? (the issue is if there ...