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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37
votes
6answers
3k 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 ...
28
votes
1answer
5k 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
582 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 ...
13
votes
4answers
619 views

On the invertibility of adjacency matrix

Which are the sufficient and necessary conditions for an undirected graph with no self edges (i.e. no loop of length 1) to have an invertible adjacency matrix? I recall that in this case the ...
11
votes
5answers
3k views

Spectrum of adjacency matrix of complete graph

Fooling around in matlab, I did an eigenvalue decomposition of the adjacency matrix of $K_5$. ...
11
votes
1answer
953 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 ...
9
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?
9
votes
0answers
187 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 ...
8
votes
1answer
215 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
226 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
2k 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
621 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
466 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 ...
6
votes
1answer
116 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 ...
6
votes
1answer
283 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, ...
6
votes
0answers
246 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, ...
5
votes
2answers
644 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
503 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
votes
1answer
305 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
123 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
136 views

Which graphs do have invertible adjacency matrices?

I would like to know if there is any class of graphs for which the adjacency matrices are invertible. At this moment I am aware of only the class of graphs $n K_2$ which is the disjoint union of $n$ ...
5
votes
1answer
142 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
0answers
105 views

Theoretical link between the graph diffusion/heat kernel and spectral clustering

The graph diffusion kernel of a graph is the exponential of its Laplacian $\exp(-\beta L)$ (or a similar expression depending on how you define the kernel). If you have labels on some vertices, you ...
5
votes
1answer
124 views

Bounding 2nd-smallest eigenvalue of the Laplacian of the binary tree

I am reading on my own the notes of this lecture series from 2012: http://www.cs.yale.edu/homes/spielman/561/2012/lect04-12.pdf. In section 4.7.2 (page 8) it's mentioned that we can prove a lower ...
5
votes
0answers
162 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
2answers
78 views

Determinant of $ n \times n$ matrix and its characteristic polynomial.

Suppose, $M_4, M_5,..M_n$ is as follows then determinant and characteristic polynomial of $M_n$. $M_4=\left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 ...
4
votes
1answer
57 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
85 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
105 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
295 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
144 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
260 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
28 views

What does the spectrum of the adjacency matrix of a graph tell you? [duplicate]

I am trying to search for an answer to the following question and I cannot find a straightforward answer. What does the spectrum of the adjacency matrix (set of eigenvalues and their multiplicities) ...
4
votes
0answers
28 views

Multiplicity of 0 eigenvalue of directed graph Laplacian matrix

I am looking for a result (if it exists) for directed graphs relating the multiplicity of 0 eigenvalues of the directed Laplacian matrix. Consider a directed graph ...
4
votes
0answers
48 views

Lower bound for spectral gap for graph on $n$ vertices

Let $G = (V,E)$ be a graph on the vertex set $V$ with edges $E$. Let $A$ be the adjacency matrix for $G$ (so $A_{ij} = 1$ if vertices $v_i$ and $v_j$ are connected by an edge), and $D$ be the ...
4
votes
0answers
116 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
263 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
82 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
72 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
543 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
1answer
79 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
2answers
364 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
151 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
3k 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
1answer
46 views

Trace of hadamard product of adjacency matrices based on eigenvalues

Let $A$ be the adjacency matrix of the simple graph $G_n$. Is there any formula for $\text{trace}(A^3 \circ A^2)$ based on eigenvalues of $A$?
3
votes
1answer
170 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 ...
3
votes
1answer
139 views

Construction of a Strongly Regular Graph which has regular Neighbourhood graphs in all iteration.

Notation and Definition: $G$ is a Strongly Regular Graph (not complete or a cycle) and is denoted by $\mathrm{SRG}(n,r, \lambda, \mu)$ if it has the following properties: Every two adjacent ...
3
votes
1answer
29 views

How to perturb an adjacecny matrix in order to have the highest increase in spectral radius?

Let's suppose I have a generic directed graph $G$ and it's adjacency matrix $A$. I can add an arc wherever I want in the graph. (i.e. perturb the matrix A changing a single 0 into a 1). Where should ...
3
votes
3answers
668 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
44 views

sum of the modulus eigenvalues of a matrix A >= B.

Kindly help me to prove/disprove the following statement. Let $A$ be a symmetric matrix of order $n \times n$ with all the diagonal entry equal to $0$, and other non-diagonal entry equal to $k$ ...