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### Intuitively, what does a graph Laplacian represent?

How to understand the Graph Laplacian (3-steps recipe for the impatients) read the answer here by Muni Pydi. This is essentially a concentrate of a comprehensive article, which is very nice and well-...
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Here is a construction. It elaborates from perturbation analysis of eigenvalues. However it starts from the situation of a non-simple eigenvalue. So, let me start with the standard self-adjoint $L_0=-\... • 48.5k 18 votes ### Reasons for difficulty of Graph Isomorphism and why Johnson graphs are important? Johnson graphs do not cause difficulty to existing programs. Actually they are rather easy; nauty can handle them up to tens of millions of vertices, and so can other programs such as Traces and Bliss.... • 34.8k 18 votes ### Rigorous justification for this formal solution to$f(x+1)+f(x)=g(x)$A direct and precise way to arrive at your answer is to appeal to the theory of Fourier transforms of distributions. If I define the Fourier transform as $$F(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\... • 155k 17 votes Accepted ### Are these three different notions of a graph Laplacian? These are usually known as the Laplacian, the normalized Laplacian and the unsigned Laplaian. All three are positive semidefinite. If the graph is regular, they all provide the same information. If ... • 11.8k 17 votes Accepted ### Differentiability of eigenvalues of positive-definite symmetric matrices In the open subset of M_n(\mathbb{R}) where the \lambda_i are distinct, they are C^{\infty} functions: this follows from the implicit function theorem. On the other hand, when some eigenvalue ... • 34.8k 17 votes ### Intuitively, what does a graph Laplacian represent? This is just a long comment, adding to the excellent answers above. There is a great article from László Lovász "Discrete and Continuous: Two sides of the same?", written around 2000 (https:/... • 6,515 16 votes ### Is there a continuous analogue of Ramanujan graphs? In fact, the original motivation behind Lubotzky--Phillips--Sarnak's construction of Ramanujan graphs was in analogy with modular curves Y(N)=\mathbb H^2/\Gamma(N) for the principal congruence ... • 17.7k 16 votes ### Differentiability of eigenvalues of positive-definite symmetric matrices The keyword is the Cartan decomposition in the theory of symmetric spaces. In short, when an eigenvalue is simple (its multiplicity is 1) it is locally an analytic function. But when the ... • 2,600 16 votes ### Spectral symmetry of a certain structured matrix An equivalent trick : Let J:= \operatorname{diag}(1,i,-1,-i). Then J^*AJ=iB where B is real and skew-symmetric. Hence the spectrum of iB (thus that of A) comes by pairs \pm\lambda. • 48.5k 15 votes Accepted ### An orbit of symmetric polynomials Suppose f(x,y,z) is symmetric (in the following, symmetric tout court always means "symmetric w.r.to the three variables (x,y,z)") . Then \mathcal{L}f(x,y,z):=(x+y)(x+z)f(x-1,y,z)-x^2f(x,y,z) ... • 52.2k 15 votes Accepted ### Imaginary eigenvalues Define the unitary and Hermitian matrices$$U=\left( \begin{array}{cccc} 0 & 0 & -i & 0 \\ 0 & 0 & 0 & -i \\ i & 0 & 0 & 0 \\ 0 & i & 0 & 0 \\ \end{... • 155k 14 votes Accepted ### Bounded operator on a normed space with empty spectrum Take an operator on a Banach space whose image is dense, whose spectrum is$\{0\}$but that has no kernel, for example $$T(f)(x)=\int_x^1f(y)dy$$ acting on$H:=L^2([0,1])$. Then its restriction to ... 14 votes Accepted ### Eigenvalues of Laplace-Beltrami on half sphere Using symmetry, you can extend any Dirichlet eigenfunction on the upper half-sphere to the entire sphere$\mathbb{S}^{N-1}$. Therefore, the spectrum of the upper hemisphere is a subset of the spectrum ... • 4,441 14 votes ### Algebraic graph invariant$\mu(G)\$ which links Four-Color-Theorem with Schrödinger operators: further topological characterizations of graphs?

Perhaps I can contribute to the history part the question, since I was quite close to the Institut Fourier at that time and was very interested in their work (I am a physicist). Grenoble now has ...

### Intuitively, what does a graph Laplacian represent?

This is not really about the connection with graph theory, a topic I am rather ignorant of, but rather the connection to continuum notions, all of which I learned from this paper. Consider a ...
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### Spectrum of matrix involving quantum harmonic oscillator

The Hamiltonian $$H=\begin{pmatrix} \alpha^2+a^\ast a&\alpha a+\beta a^\ast\\ \alpha a^\ast+\beta a&\beta^2+a^\ast a \end{pmatrix}$$ is known in the physics literature as the anisotropic Rabi ...
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