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Results tagged with spectral-graph-theory
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user 11260
Questions related to the spectrum of graphs, defined using one of the possible variants of the discrete Laplace operator or Laplacian matrix. See https://en.wikipedia.org/wiki/Discrete_Laplace_operator
10
votes
regular graphs with the smallest eigenvalue -2?
Connected regular graphs with smallest eigenvalue at least $−2$ are either a line graph, a cocktail party graph, or the number of vertices is at most 28.
P. J. Cameron, J. M. Goethals, J. J. Seidel a …
- 188k
1
vote
Accepted
Are the eigenvalues of the 1D lattice with random weights known?
I presume this will depend on the connectivity of the 1D lattice. Let me consider the simple case of nearest neighbor connections, when the adjacency matrix $A$ is tridiagonal with the same values on …
- 188k
1
vote
Is there a version of Weyl's law for graph Laplacians?
For $N_\lambda$ denoting the number of eigenvalues less than $\lambda$, Weyl's law gives the asymptotics of $N_\lambda$ as $\lambda$ tends to infinity. The usual approach to establish this asymptotics …
- 188k
7
votes
Accepted
exact definition of Fiedler vector
The concept of a Fiedler vector is defined for graphs that consist of one single connected component. Since the number of zero eigenvalues counts the number of connected components, the second largest …
- 188k
3
votes
What is the state of the art on triangle-free strongly regular graphs?
an update of Biggs's list (complete up to 1300 vertices) is maintained here by Andries Brouwer.
for a list of open problems and research directions, a good starting point could be Matan Ziv-Av's rece …
- 188k
3
votes
Fiedler vector, what else?
The Fiedler vector refers to the second smallest eigenvalue, here is a study of
The third smallest eigenvalue of the Laplacian matrix (2001).
The relationship between the third smallest eigenvalue of …
- 188k
3
votes
Accepted
Relation between Kirchhoff's Circuital law and Matrix tree Theorem
Chapter II (pages 12 and following) of Combinatorics of Electrical Networks gives a linear algebra derivation of Kirchhoff's theorem from the circuit laws of Ohm and Kirchhoff.
- 188k
2
votes
What happens to eigenvalues when edges are removed?
This might be helpful, from Eigenvalues and structures of graphs
- 188k
9
votes
Accepted
graph signal processing
"I am looking for some simple concrete examples of the ways in which
real problems go through graph signal processing and how graph Fourier
transforms are obtained."
• A concrete example of a …
- 188k
55
votes
Accepted
What are good mathematical models for spider webs?
In response to the second question (which I interpret as asking for math models of spider webs as they appear in Nature): There exist several distinct types of spider webs. The most common type, the o …
- 188k
4
votes
Multiplicity of the smallest non-zero Laplacian eigenvalue for tree graphs
The multiplicity of Laplacian eigenvalues of tree graphs is studied in arXiv:1907.11482. If $\Delta$ is the maximal degree of the graph, and $\Delta\geq 2$, then the multiplicity $m_2$ of $\lambda_2$ …
- 188k