Convex optimization of graph laplacian eigenvalues

  • 1 Answer.
    Assuming you are talking about the Laplacian matrix of a simple (undirected) graph, you were right: it never has negative eigenvalues.

  • What are the eigenvalues of Laplacian of complete bipartite graph?

    The Laplacian matrix of a complete bipartite graph Km,n has eigenvalues n + m, n, m, and 0; with multiplicity 1, m − 1, n − 1 and 1 respectively.
    A complete bipartite graph Km,n has mn−1 nm−1 spanning trees.
    A complete bipartite graph Km,n has a maximum matching of size min{m,n}..

  • What are the eigenvalues of the Laplacian of a complete graph?

    For a complete graph on n vertices, all the eigenvalues except the first equal n.
    The eigenvalues of the Laplacian of a graph with n vertices are always less than or equal to n, this says the complete graph has the largest possible eigenvalue..

  • What are the eigenvalues of the Laplacian of a completely connected graph?

    For the complete graph Kn on n vertices, the eigenvalues are 0 and n/(n − 1) (with multiplicity n − 1).
    Example 1.2..

  • What do the eigenvalues of Laplacian tell us?

    Spectral graph theory, looking at the eigenvalues of the graph Laplacian, can tell us not just whether a graph is connected, but also how well it's connected.
    The graph Laplacian is the matrix L = D – A where D is the diagonal matrix whose entries are the degrees of each node and A is the adjacency matrix..

  • What is the eigenvalue of the Laplacian?

    All eigenvalues of the normalized symmetric Laplacian satisfy 0 = μ0 ≤ … ≤ μn−1 ≤ 2.
    These eigenvalues (known as the spectrum of the normalized Laplacian) relate well to other graph invariants for general graphs..

  • The Laplacian L=Du221.
    1. A works well for the regular graphs but the Normalised laplacian ℒ=D−1/
    2. LD1/2=D−1/2(Du221
    3. A)D1/2=Iu221
    4. D−1/
    5. AD1/2 not only works well for regular but also irregular graphs
We consider the problem of choosing the edge weights of an undirected graph so as to maximize or minimize some function of the eigenvalues of the associated.

Are vertices eigenvectors of the graph Laplacian?

For a regular polyhedron (or polygon) centered at the origin, the coordinates of the vertices are eigenvectors of the graph Laplacian for the skeleton of that polyhedron (or polygon) associated with the first (non-trivial) eigenvalue

In this paper, we generalize this relationship

How do you find the optimal eigenvalue of a graph?

Find edge weights that maximize the algebraic connectivity of the graph (i

e

, the smallest positive eigenvalue of its Laplacian matrix)

The optimal value is called the absolute algebraic connectivity by Fielder

• Minimum total effective resistance

Find edge weights that minimize the total effective resistance of the graph

What if the second eigenvalue is not close to 0?

If the second eigenvalue is not close to 0, the graph is too well connected to simply remove a few edges without compromising the originial structure of the graph

The de nition of Cheeger's constant and the statement of Cheeger's inequality will prove our claim


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