Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [graph-laplacian]

A simple graph has a symmetric matrix L = D - A associated with it called the Laplacian matrix, where D is the diagonal matrix of degrees and A is the adjacency matrix, often studied for its spectrum (eigenvalues).

A simple graph has a symmetric matrix L = D - A associated with it called the Laplacian matrix, where D is the diagonal matrix of degrees and A is the adjacency matrix, often studied for its spectrum (eigenvalues).

The Laplacian matrix for simple graphs is positive semi-definite. Variations using indegree or outdegree can be defined for directed graphs.

293 questions
21
votes
2 answers

What can we say about the graph when many eigenvalues of the Laplacian are equal to 1?

The Laplacian of the graph has all the eigenvalues real and non-negative, the smallest being 0. I have a graph where the second smallest eigenvalue (the so called algebraic connectivity) is equal to $1$. In fact, the multiplicity of this eigenvalue…
Peaceful
  • 581
21
votes
3 answers

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

Why Laplacian matrix needs normalization and how come the sqrt-power of degree matrix? The symmetric normalized Laplacian matrix is defined as $$\ L = D^{1/2}AD^{-1/2}$$ where L is Laplacian matrix, A is adjacent matrix. Element $A_{ij}$ represents…
John Buger
  • 213
19
votes
3 answers

Meaning of the inverse of the Laplacian matrix

Given an undirected graph $G = (V,E)$, let $\bf A$ and $\cal L_{\bf A}$ denote its adjacency matrix and its Laplacian matrix, respectively. $\cal L_{\bf A}(i,i)$ is the degree of vertex ${{\bf{v}}_i}$, and $\mathcal L_{\bf A}(i,j) = -1$ if vertices…
Tony
  • 5,716
18
votes
3 answers

Why is second smallest eigenvalue and the corresponding eigenvector used to partition a graph?

In spectral graph partition theory, the eigenvector $v_2$ (also called Fiedler vector) corresponding to the second smallest eigenvalue $\lambda_2$ (also known as Fiedler eigenvalue, which actually also defines the algebraic connectivity of a graph)…
guanglei
  • 863
15
votes
1 answer

Writing the Laplacian matrix of directed graphs as a product

The Laplacian matrix of an undirected graph can be written as $M^T M$ with $M$ being the incidence matrix of the graph. This makes the (otherwise tedious) proof of Kirchhoff's theorem into a beautiful application of the Cauchy-Binet formula (and…
Gadi A
  • 19,839
15
votes
3 answers

Geometric intuition of graph Laplacian matrices

I am reading about Laplacian matrices for the first time and struggling to gain intuition as to why they are so useful. Could anyone provide insight as to the geometric significance of the Laplacian of a graph? For example, why are the eigenvectors…
T. Silver
  • 345
12
votes
1 answer

Laplacian of a directed weighted graph

I know that for a simple undirected graph $\mathcal{G}(V,E) $ the Laplacian matrix $L$ is defined as: $$ L:=D-A$$ where $D$ is the degree diagonal matrix and $A$ is the adjacency matrix of $\mathcal{G}$. My question is: how can I define the…
Leonardo Massai
  • 1,943
11
votes
1 answer

Why is a graph Laplacian matrix positive semidefinite?

Why is a graph's Laplacian matrix positive semidefinite? Can anyone provide an intuitive explanation and a proof?
user1559897
  • 1,855
11
votes
3 answers

What does the value of eigenvectors of a graph Laplacian matrix mean?

I know that the eigenvectors of a Laplacian matrix of a graph are so important. They show the locality over the graph (as I know). But whatever I've read about an eigenvector of Laplacian graph is about the smoothness of the eigenvector and the…
user137927
  • 590
10
votes
3 answers

Spanning Trees of the Complete Graph minus an edge

I am studying Problem 43, Chapter 10 from A Walk Through Combinatorics by Miklos Bona, which reads... Let $A$ be the graph obtained from $K_{n}$ by deleting an edge. Find a formula for the number of spanning trees of $A$. So how I approached this…
Scott
  • 363
9
votes
1 answer

Why do we use the Laplacian matrix in Spectral Clustering?

When we perform spectral clustering, given a similarity matrix $S$, we define the Laplacian matrix $L$ (normalized or unnormalized). Then, we do eigenvalue decomposition on $L$ and get its eigenvector matrix. Why do we do eigenvalue decomposition on…
big_shw
  • 93
8
votes
1 answer

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? I'm specially interested to know if there's any…
SKV
  • 254
8
votes
1 answer

Perron-Frobenius Theorem and Graph Laplacians

How can the Perron-Frobenius theorem be used to show that for a connected graph, there is a simple eigenvector that is (i) real and (ii) smallest in magnitude and (iii) has an associated eigenvector that is positive? The graph Laplacian is given as…
1yen
  • 257
8
votes
1 answer

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. Considering the spectral function $D+\rho A$ over…
alancalvitti
  • 3,440
8
votes
0 answers

Finding the odometer function of an abelian sandpile

As a computer scientist and "armchair" mathematician, I'm trying to replicate the images found here of Abelian sandpiles on an infinite square lattice, where the initial configuration is $n$ chips on a single vertex and all others empty. My best…
mgold
  • 293
1
2 3
…
19 20