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 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 ...
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3 views

What Laplacian should we use for spectral clustering?

The second eigenvector of the normalized Laplacian $I-D^{-1}W$ or the symmetric normalized Laplacian $I-D^{-1/2}WD^{-1/2}$ can be used to approximate a minmizer of the normalized cut problem. Which ...
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51 views
+50

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 ...
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1answer
26 views

Optimal partitioning of a planar graph

Consider a planar graph, where each node is associated with a weight. I would like to partition the graph such that the sum of the node weights in each group satisfy a minimum requirement. However, I ...
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15 views

Spectrum of the circulant graph

How to prove that the eigenvalue of cycle $C_n=\lambda_r=2 cos(2\pi r/n)$?where $r=0,1,...n-1$, which is proved for the circulant matrix with first row $(v_0=0,v_1=1,v_2=0, ...v_{n-2}=0,v_{n-1}=1)$, ...
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40 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 ...
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18 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 ...
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2answers
61 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 ...
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48 views

Who are the mathematicians in US who are working on expander graphs right now?

I am familiar with only the "big" names doing this research like Gharan, Nikhil Srivastava, Dan Spielman, Jean Bourgain, Luca Trevisan, Elina Fuchs, Peter Sarnak , Amin Saberi and Terence Tao. I would ...
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0answers
20 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 ...
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27 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 ...
3
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1answer
125 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 ...
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1answer
41 views

Computing eigenvalue of the adjacency matrix of a path

Let $A\in \{0,1\}^{n \times n}$ be the adjacency matrix of a path of length $n$, i.e. having ones on the two off-diagonals, and zeros elsewhere. How does one compute the eigenvalues of this? I know ...
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0answers
26 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 ...
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2answers
33 views

Confusion between spectral radius of matrix and spectral radius of the operator

The adjacency matrix $A(G)$ of an infinite undirected graph $G$ is considered as a bounded self-adjoint linear operator $A$ on the Hilbert Space $l^2(G)$ (last section of ...
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1answer
57 views

Characteristic polynomial of a block matrix

Let $A$ be an $n\times n$ symmetric matrix whose diagonal are is covered by zero blocks (square, but not of a fixed size) and all other entries are $1$ (one). How can I find its Characteristic ...
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36 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 ...
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0answers
15 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, ...
3
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15 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 ...
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1answer
33 views

Relation between edge expansion of graph and sparsity

I was going through the lectures of Graph Partitioning and Expanders - Stanford Online. In lecture 1, near the end of page 5, I came across this inequality for regular graphs: $$\phi(S) \leq h(S) \leq ...
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11 views

The image of a a vector in the edge space when multiplied by it's incidence matrix.

Consider a graph $G=(V,E)$ and it's incidence matrix $M$. Let $\textbf{x}$ be the characteristic vector for a standard basis vector in $\mathcal{E}$ (a vector corresponding to the one element edges ...
3
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0answers
36 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 ...
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3 views

PageRank metaphor as suspended unlimited capacity pools

What do you think about my metaphor of the centrality measure PageRank? PageRank is an algorithm for evaluating node centrality: it's a function $f:G \to R^n$ where $n$ is the number of nodes in the ...
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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 ...
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1answer
24 views

Construction of graph Laplacian

I have a weighted undirected graph, and all the edge-weights are non-negative. According to the definition of the graph Laplacian matrix, $L=D-W$. In literature, I found that $D$ is known as degree ...
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0answers
42 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 ...
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1answer
37 views

spectrum of complete p-partite graphs

I need to determine the spectrum of the complete p-partite graph ( in which each partite set has m vertices) using the complement. How can i show this? I know the spectrum of the adjacency matrix of ...
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0answers
36 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 $$ ...
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13 views

Algebraic connectivity (normalized Laplacian): does the non-weighted instance define the upper bound for the weighted graphs with the same topology?

I am not a mathematician but I use a lot of maths. I came across some empirical evidence that the algebraic connectivity (2nd smallest eigenvalue of the random walk normalized laplacian matrix) of ...
2
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1answer
86 views

Eigen value and Regular Graph (not Strongly Regular graph).

$A,B$ are 2 adjacency matrices of $d$ Regular graphs(not Strongly Regular graphs). I would like to know- 1.Results/ information related to Eigen values of A,B. There is a formula for ...
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1answer
45 views

Triangles incident to a node i

I'm trying to use some fragment-based measures for a network. Given an adjacency matrix representing a (large) network how do you calculate the number of triangles that are incident to every node i? ...
2
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1answer
73 views

Eigen value of principal submatrix.

I was studying "interlacing property" and trying to find out the below fact- $A$ is an adjacency matrix of a $r$ regular graph $G$. $u,v \in G $;$u,v$ are not similar vertices. $B$ is the ...
3
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53 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 ...
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70 views

Graph Combinatorics: How many such Graphs are there?

How many $4$-regular graphs exist on $8$ vertices? I found that such a graph can't be disconnectd since if so, then graph can be written as disjoint union of atleast two graphs. $4$ regularity ...
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25 views

How to show a random matrix has large spectral gap?

If I know $Y$ is a random d-regular bipartite graph (tanner code in coding theory), can I show $Y^TY$ has large spectral gap with high probability? More specifically: If I know $Y=AX \in ...
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23 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
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1answer
51 views

Icosahedral Graph

Let $Γ$ be a graph cospectral with the icosahedral graph having spectrum $\{[5]^1,[\sqrt{5}]^3, [-1]^5,[-\sqrt{5}]^3\}$. I have shown that Γ has 12 vertices, 30 edges, regular with each vertex having ...
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0answers
29 views

How to find spectral radius of ${0,1}$ and ${0,1,-1}$ matrices?

[this is kind of a continuation of this question ] It seems that the following is true, Among $n=3$ dimension symmetric matrices over $\{0,1\}$ which have $d=7$ ones the maximum spectral radius is ...
0
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1answer
51 views

What is known about optimization of spectral properties of matrices over finite fields?

[I am solving the characteristic polynomial over complex numbers but since the matrices are symmetric all eigenvalues are real] Like for symmetric $d-$regular matrices over 0/1 or 0/1/-1 what are ...
4
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1answer
51 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 ...
2
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1answer
86 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 ...
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0answers
40 views

Intuition behind spectral radius of a graph

Suppose that I have a graph G, along with its respective adjacency matrix A. The definition of how one can compute the spectral radius of this graph is not hard to grasp, but I was wondering about the ...
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25 views

Graph Centrality: spectral techniques

What is the difference between: normalizing the row of an adjacency matrix and taking the right eigenvector normalizing the row of an adjacency matrix and taking the left eigenvector ...
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95 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 = ...
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1answer
74 views

A curious way to write the eigenvectors of the Boolean hypercube

It seems that one can write the eigenvectors of the hypercube $\{ \pm 1\}^n$ as the functions, $\{ \chi_S \}_{S \subseteq [n] }$. And these functions $\chi_S$ are defined on the vertices $x \in \{ \pm ...
2
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1answer
59 views

Efficiently compute the eigenvectors of the Laplacian of a symmetric positive matrix

I am working with a matrix A relatively large (200k x 200k), and I want to compute the eigenvectors of the Laplacian: $L = D - A^2$, where $A$ is symmetric. I don't need all eigenvectors, just a few ...
0
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1answer
23 views

What conditions on a graph $G$ allow it to be uniquely determined by the spectrum of $A(G)$?

What conditions on an undirected graph $G$ allow it to be uniquely determined by the spectrum of its adjacency matrix $A(G)$? Very simple examples show that one needs connectivity, and I imagine ...
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130 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 ...
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23 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 ...
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73 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 $d$? ...