For questions related to the study of properties of a graph in relationship to the spectral properties of some associated matrix.

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34
votes
6answers
2k views

Motivation for spectral graph theory.

Why do we care about eigenvalues of graphs? Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so ...
1
vote
0answers
58 views

Weighted undirected graphs, complex Laplacian, complex eigenvalues & spectral clusering

I am rather puzzled and confused, I have been trying to get a clear understanding of how would spectral clustering work for an undirected weighted graph, I have used the normalized Laplacian, but I ...
27
votes
1answer
4k views

What do the eigenvectors of an adjacency matrix tell us?

The principal eigenvector of the adjacency matrix of a graph gives us some notion of vertex centrality. What do the second, third, etc. eigenvectors tell us? Motivation: A standard information ...
11
votes
5answers
2k views

Spectrum of adjacency matrix of complete graph

Fooling around in matlab, I did an eigenvalue decomposition of the adjacency matrix of $K_5$. ...
3
votes
2answers
136 views

Are the eigenvectors of vertex transitive graphs bounded

For a connected and regular graph $G$ with degree $ d $ at each vertex and adjacency matrix $A$, the normalized Laplacian of $G$ is defined as $L = I-\frac{1}{d}M$. Let $\psi$ be an eigenvector of $L$ ...
4
votes
1answer
225 views

Questions on fractional Laplacian graph spectra

Both the signed ($D-A$) and unsigned ($D+A$) Laplacian are of interest in spectral graph theory, see eg Cvetkovic: Bibliography on the signless Laplacian eigenvalues: first one hundred references. ...
3
votes
2answers
215 views

How adjacency matrix shows that the graph have no cycles?

Let $G$ a directed graph and $A$ the corresponding adjacency matrix. Let denote the identity matrix with $I$. I've read in a wikipedia article, that the following statement is true. Question. Is it ...
4
votes
0answers
175 views

What can we say about two graphs if they have similar adjacency matrices?

Suppose we have two (finite, simple, undirected) graphs, what can we say about these graphs if they have similar adjacency matrices? Observations to begin with: If $G_1$ and $G_2$ are isomorphic, ...
2
votes
1answer
401 views

Connectedness of a regular graph and the multiplicity of its eigenvalue

Suppose $X$ is a $k$-regular graph with adjacency matrix $A$. I wish to show that if $k$ has multiplicity $1$ as an eigenvalue of $A$ then $X$ is connected. By way of contradiction I assume that X is ...
0
votes
1answer
46 views

What is known about optimization of spectral properties of matrices over finite fields?

[I am solving the characteristic polynomial over complex numbers but since the matrices are symmetric all eigenvalues are real] Like for symmetric $d-$regular matrices over 0/1 or 0/1/-1 what are ...