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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9
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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 ...
7
votes
0answers
464 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
0answers
245 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
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
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
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
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 $k$ ...
3
votes
0answers
36 views

Possible lower Bound of radius of a graph

First of all I would like to ask people to forgive me because the question that I am about to ask is based on results of a study that I was involved in this year. The project was about designing ...
3
votes
0answers
38 views

What do we know about inverses of matrices which are “like” Laplacians of graphs?

Consider the Laplacian $L$ of a bipartite graph. Is there any generic understanding we have about what $1/(z-L)$ looks like? [say $z > \lambda_\max(L)$)] You can consider variations of $L$ like ...
3
votes
0answers
27 views

How networks with high largest eigenvalues are more robust?

In the literature, it sometimes indicates that network with high value of largest eigenvalue (either adjacency matrix or its Laplacian counterpart) are more robust. Robustness here is relevant to ...
3
votes
0answers
71 views

Is there an easy way to realize a graph (i.e. get adjacency matrix) from a fundamental cut-set or loop matrix?

I am looking to realize a graph (i.e. write down its adjacency or incidence matrix) given a fundamental cut-set matrix or loop matrix (with respect to an arbitrary spanning tree). Is there some ...
3
votes
0answers
179 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
70 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 $\phi(G)$ ...
2
votes
0answers
7 views

Can a graph be recovered from its Bonacich centrality vector?

Let $A$ be the adjacency matrix of a directed graph with $n$ vertices and spectral radius $\lambda$. Let $I$ be the $n \times n$ identity matrix and let $e \in \mathbb{R}^n$ be the vector of 1's. For ...
2
votes
0answers
199 views

Is each edge interpreted like a $2$-cycle?

Let $a_k$ a an eigenvalue of the adjacency matrix $A$ of a planar cubic graph with $n$ vertices. For the returning paths without backtracking we get the generating function of ...
2
votes
0answers
24 views

Integral roots of a circulant matrix

When does the circulant matrix have only integral roots? For example: adjacency matrix for $K_n$ has all the roots integral which is circulant, but in case of Cycle on $n>3$ it is circulant but it ...
2
votes
0answers
68 views

Eigenvalue of Block matrix: Adjacency of complete bipartite Graph

Let $A\in \{0,1\}^{mn \times mn}$ be the adjacency matrix of a complete bipartite graph with $m$ and $n$ vertices each, i.e. let $A$ be the matrix consisting of two blocks $A_1\in \{0,1\}^{m \times ...
2
votes
0answers
45 views

When do a Regular graph has an odd eigenvalue?

Merely looking at adjacency matrix of a regular graph, without explicit calculation, can we decide that graph will have an odd eigenvalue or not? If regularity is odd, we are sure that it will be an ...
2
votes
0answers
120 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 = ...
2
votes
0answers
42 views

About the topology of a $d$-regular tree

What is the proof that the infinite $d$-regular tree is an universal covering space for any $d$-regular graph? Is it true that the infinite $d$-regular tree is a Ramanujan graph? (any easy way to see ...
2
votes
0answers
35 views

Spectral radius of a time-varying matrix with strictly positive increment

Consider a time varying non-negative matrix $A(t)$ and its spectral radius $\rho(A(t))$ where $t$ denotes the time. If $A(t)$ changes over time with each time a random element in $A(t)$ is being ...
2
votes
0answers
154 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 & ...
2
votes
0answers
112 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 ...
2
votes
0answers
59 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 ...
2
votes
0answers
95 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 $A(L_{sym}) = I - L_{sym}$ ...
2
votes
0answers
349 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 ...
2
votes
0answers
52 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 ...
2
votes
0answers
738 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 ...
1
vote
0answers
26 views

Spectral radius of perturbed bipartite graphs

I am looking into how perturbation(s) on a bipartite graph affect its spectrum (specifically its spectral radius or largest eigenvalue). Actually I'm not exactly looking into bipartite but the ...
1
vote
0answers
45 views

Semigroup of matrices and expander Cayley graphs

I am interested in proving or disproving that certain Cayley graphs are expander. Let $S$ be the multiplicative semigroup of matrices generated by $A = \left( \begin{array}{cc} a & b \\ 0 & ...
1
vote
0answers
21 views

Probability having a path of length less than a fixed number

A graph $G(V, E)$ is given. For a random pair of nodes $e_1, e_2 \in V$, what is the chance/probability of having a path of size less than $k$ (a fixed number) between $e_1$ and $e_2$ (let's assume ...
1
vote
0answers
57 views

Finding all eigenvalues of the adjacency matrix of a simple graph

I want to find all eigenvalues of the adjacency matrix of the following graph(Graph spectrum), where $G$ and $H$ are complete graphs with $n$ and $m$ vertices, respectively, for positive integers $n,m ...
1
vote
0answers
36 views

Graph having bounded degree

A graph is said to have bounded degree if there exists $N \in \mathbb{N}$ such that, for every $x \in V$, one has $\sum\limits_{y \in V} A_{x,y} \le N$. Show that, in this case, for any $f \in ...
1
vote
0answers
37 views

A question about minimizing the $\lambda_{max}$ over a set of diagonal perturbations

Say I have an off-diagonal symmetric $0,1,-1$ entry matrix $B$ and a set of $2k$ diagonal matrices, $D_{11}, D_{12}, D_{21}, D_{22},..,D_{k1},D_{k2}$. (if it helps you can assume that $(1)$ all the ...
1
vote
0answers
68 views

Eigenvectors of graph laplacian

Let $L$ be the laplacian matrix of a graph $G$, i.e. $L = D - A$, where $D$ is the degree matrix, and $A$ the adjacency matrix. Let $v_i$ be an eigenvector of $L$. Let $x,y$ be two vertices of the ...
1
vote
0answers
29 views

What is the adjacency matrix of a squared (or k^th power) d-regular graph

If $A$ is the adjacency matrix of a $d$-regular graph, then I suppose $A'$(the adjacency matrix of the squared graph) should be $A^2 + A - dI$ (to remove self-loops). What about higher powers? How do ...
1
vote
0answers
43 views

Highest eigenvalues/vectors of graph laplacian

In page 8 of the paper Laplacian Eigenmaps for dimensionality reduction and Data Representation it reads: Standard methods show that the solution is provided by the matrix of eigenvectors ...
1
vote
0answers
17 views

Size of intersection of balls on non-ameanable graphs

Let $G$ be a vertex-transitive non-ameanable graph and let $B(x,n)$ be the ball of radius $n$ centered on the vertex $x$. I am interested in estimates on the cardinality of the following set, ...
1
vote
0answers
55 views

Interpretations of a weighted adjacency matrix's eigenvectors and eigenvalues?

Suppose that I have weighted undirected graph $G$, and the corresponding adjacency matrix which is a symmetric matrix $A$. Suppose that the edge between node $i$ and $j$ has weight $w_{ij}$, then $$ ...
1
vote
0answers
26 views

Expected number of leaf nodes in some theoretical graph models

If a leaf node of a graph refers to a node having the degree of 1, how can one compute the expected number of leaf nodes of: (A) a random graph (e.g., Erdos-Renyi graph), (B) a small-world graph ...
1
vote
0answers
32 views

Adjacency vs Laplacian matrix

There are different ways to represent a graph but adjacency and laplacian matrices are the two most powerful ones having various properties. Recently, a student asked me when exactly we should use ...
1
vote
0answers
48 views

Spectral gaps of common graphs

I'm looking for the spectral gap of common graphs (alternatively, the mixing time of a (lazy) random walk on these graphs). Asymptotic values are fine. Assume that every node has a sufficient number ...
1
vote
0answers
63 views

Properties of non-negative non-symmetric square matrices

I've done some searching but couldn't get much from the web. I am looking for some pointers regarding the properties of non-negative non-symmetric square matrices. The elements within the matrix are ...
1
vote
0answers
96 views

Is there such a notion of “expansion” in groups?

Given a subset of elements of a finite group $G$, I would like it to be such that the set of all distinct words (as elements of $G$) that can be formed from this set is exponentially large in the size ...
1
vote
0answers
123 views

Weighted undirected graphs, complex Laplacian, complex eigenvalues & spectral clusering

I am rather puzzled and confused, I have been trying to get a clear understanding of how would spectral clustering work for an undirected weighted graph, I have used the normalized Laplacian, but I ...
1
vote
0answers
43 views

Fast Cholesky Factrorization for Tree Laplacians

Suppose $T_1$ and $T_2$ represent two Laplacian matrices of two spanning trees of $n$ vertices. Since the Cholesky factorization needs $O(n)$ time for each $T_i\ (i=1,2)$ due to the tree structure, ...