# 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).

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### How to understand the potential energy cost function based on cotan Laplacian operator in 3D mesh deformation

Suppose I have a source triangle mesh surface $S^0 = (V,E)$. $V,E$ are sets of $n$ vertices and $m$ edges of the mesh, respectively. $S^1,S^2,S^3, \ldots, S^l$ is denoted as a sequence of meshes after ...
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### Calculating the diagonal entry of the adjugate of the Laplacian matrix of a complete graph.

I am trying to directly calculate the adjugate of the Laplacian of the complete graph from which I know is given by $\tau(G) = n^{n-2}$ where $\tau(G)$ is the number of spanning trees of $K_n$. I ...
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### Eigenvalue corresponding to the highest eigenvalue multiplicity of normalized Laplacian spectrum

Suppose we have an undirected weighted graph (more specifically, a scale-free network). When I plot its normalized graph Laplacian spectrum, I get something like this: Normalized Laplacian Spectrum, ...
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### What does the notation subscript *L and a circle with a dot in it represent?

I was recently reading this arxiv paper, in the context of Machine Learning, and I was wondering what the notation $\odot$ and $\square^\ast_L$ are supposed to mean under section 3. I had read ...
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### How can the relationship between two nodes be captured using the Laplacian matrix?

I am new to spectral graph theory, so I apologize if there are any mistakes in my writing. If I make an error, please kindly point it out and guide me in the right direction. Now let me clearly state ...
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### Eigenvector of normalized graph Laplacian

I’m reading a paper and they make a claim about the eigenvector of a graph Laplacian that seems wrong and I wanted to check with the larger community if I’m missing something obvious. Let $G = (V,E,W)$...
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### Generalizing the sum of squared second differences to graphs?

Suppose I have a sequence of numbers $x = (x_1, x_2, \dots x_n)$. I’m interested in the sum of squared differences: $$\sum_{i=1}^{n-1} (x_{i+1} - x_i)^2.$$ I can treat the sequence as a function on ...
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### Number of Spanning Trees in $Q_n$ via isomorphism to Cartesian product of $K_2$

I'm trying to calculate (or, show) the number of spanning trees in the hypercube $Q_n$. I know I must come to $$\tau(Q_n) = \prod^{n}_{i=2} (2i)^{\begin{pmatrix} n \\ i\end{pmatrix}}$$ I thought it ...
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### Why does the condition $\sum_{x\in\Lambda}f_x=0$ imply the discrete Laplacian is invertible?

I'm studying statistical physics these days and have newly learnt the concept discrete Laplacian. For a finite graph $G$ with vertices $\Lambda\subset\mathbb{Z}^d,$ consider the discrete Laplacian ...
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### Why can a similarity matrix be used instead of a Laplace matrix when using spectral clustering methods?

When we are using spectral clustering methods, we often construct similarity matrices $S$ between data, and use the similarity matrix to derive the Laplacian matrix $L$ for further clustering. But in ...
1 vote
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### Graph diffusion: finding the steady state current

Consider a quantity $x$ diffusing on a graph according to $$\frac{\Bbb dx}{\Bbb dt}=-Lx+a,$$ where $L$ is the graph Laplacian and $a$ encodes sources and sinks. In the steady state, $x_0=L^{-1}a$. How ...
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### Determining the Laplacian for a signed graph

I was reading the paper: On the Laplacian Eigenvalues of Signed Graphs, when I had this question. When we determine the Laplacian of a signed graph, why do we take the unsigned degree matrix D, but ...
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