40 votes
Accepted

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-...
37 votes
Accepted

Spectral symmetry of a certain structured matrix

For real $a,b,c$ and imaginary $d$ the matrix $A$ has chiral symmetry, meaning it anticommutes with a matrix $X$ that squares to the identity: $$X=\left( \begin{array}{cccc} 0 & 0 & 0 & -...
36 votes

Can you hear the shape of a drum by choosing where to drum it?

Sometimes you can Since there seem to be no hard results along these lines on the literature, I decided to have a look at the two shapes in the question and see if there are some relatively ...
25 votes
Accepted

Can one hear the (topological) shape of a drum?

There are examples due to Ikeda of isospectral Lens spaces which are not homotopy equivalent. Likeliest the simplest examples are the compact connected 3-dimensional flat manifolds which are a ...
  • 62.7k
24 votes
Accepted

Why should I look at the resolvent formalism and think it is a useful tool for spectral theory?

Preliminary remark. As mentioned in the comments, I find the notion "resolvent formalism", as well as the description in the Wikipedia article, rather misleading - resolvents are not ...
23 votes
Accepted

How are eigenvalues and eigenvectors affected by adding the all-ones matrix?

This is a special case of a rank one perturbation or a rank one update, and there is plenty of work on such. See the nice 2010 lecture notes by Andre Ran.
  • 94.2k
23 votes
Accepted

Eigenfunctions of the laplacian on $\mathbb{CP}^n$

The $k$-th eigenfunctions are actually easy to describe: In $\mathbb{C}^{n+1}$ with unitary complex coordinates $z_0,z_1,\ldots,z_n$, write $Z = |z_0|^2+\cdots+|z_n|^2$. Now, for a given $k\ge0$, ...
20 votes
Accepted

What is a "Ramanujan Graph"?

Ramanujan graphs were first defined by Lubotzky, Phillips and Sarnak: http://math1.math.huji.ac.il/~alexlub/PAPERS/ramanujan%20graphs/ramanujanGraphs.pdf As you can see, they are $d$-regular and and ...
20 votes
Accepted

Why is there no symplectic version of spectral geometry?

The characteristic variety (i.e. vanishing locus of the symbol) of a symplectomorphism invariant scalar differential equation is a real projective hypersurface invariant under the group of ...
  • 23.7k
20 votes

Why is there no symplectic version of spectral geometry?

From a certain point of view the premise of the question is wrong. The study of sympletic manifolds with no additional structure is akin to differential topology rather than differential geometry. ...
  • 2,082
20 votes
Accepted

Eigenvalue pattern

The explanation is pretty simple with a suitable change of basis. Letting $$B = \begin{pmatrix} 1 & 0 & 1 & 0 \\ i & 0 & -i & 0 \\ 0 & 1 & 0 & 1 \\ 0 & i &...
19 votes
Accepted

Non real eigenvalues for elliptic equations

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....
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}^\...
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 ...
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{...
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 ...
14 votes

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 ...
  • 241
14 votes
Accepted

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 ...
13 votes

Do perfect matching(s) have signatures in the graph eigenvalues?

Blazsik, Cummings and Haemers http://arxiv.org/abs/1409.0630 recently constructed two regular cospectral graphs such that one has a perfect matching and the other does not.
13 votes

Can you hear the shape of a drum by choosing where to drum it?

As mentioned in the comments, knowing both eigenvalues and eigenfunctions gives you enough information to find the shape of the domain, so to make this problem more challenging one might ask what ...

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