All Questions
15 questions
2
votes
0
answers
79
views
Upper bound for smallest eigenvalue of infinite family of graphs
Let $\left\{G_{n}\right\}_{n=1}^{\infty}$ be a sequence of regular simple connected graphs with at least one edge such that $G_i$ is an induced sub-graph of $G_{i+1}$ and is not equal to $G_{i+1}$.
...
- 133
3
votes
1
answer
271
views
Local-Global Principle in Graph Spectrum
The question is a bit vague, but any ideas/directions will be appreciated.
Let us fix an $n$-vertex $d$-regular graph $G=(V,E)$. As I understand it, the eigenvalues of the adjacency matrix $A$ of $G$ ...
- 949
1
vote
0
answers
51
views
Relation between nullity of components to its parent graph
Let $G$ be an undirected graph and the corresponding adjacency matrix be $A$. Let $v$ be a cut-vertex of $G$. Let $G_1, G_2,\dots, G_k$ are the connected components of the induced graph $G-v$ ( the ...
- 640
5
votes
1
answer
1k
views
Intuition on Kronecker Product of a Transition Matrix
Let $T$ be a $N\times N$ transition matrix for a markov chain with $N$ states. Thus $T_{ij}$ is the probability of transition from state $i$ to state $j$ (and thus rows summing to one). Now consider ...
- 1,421
0
votes
1
answer
218
views
When is a $2$-lift of a graph connected? [closed]
Let $\ (V\ E)\ $ be a graph, i.e. $\ E\subseteq\binom V2.\ $ A $2$-lift pattern of a graph is a function $\ e:E\rightarrow\{-1\,\ 1\}.\ $ The induced 2-lift is defined as the graph $\ V\times\{-1\,\ ...
- 1,893
2
votes
1
answer
238
views
Laplacian spectrum of $2-$lifts of graphs
We know that a $2-$ lift of a graph is specified by a $\pm 1$ assignment on the edges of the graph ( given as a signing matrix) denoting which edge is to be duplicated by the identity permutation on ...
- 1,893
0
votes
1
answer
204
views
Are these particular kinds of matrices well known?
Given two positive integers $n$ and $a \leq \frac{n}{2}$ consider a $n \times n$ matrix $A$ such that,
all the diagonal entries are either $a$ or $a+1$
all the non-zero off-diagonal entries are $\pm ...
- 1,893
2
votes
1
answer
520
views
Neighborhood overlap matrix for a bipartite graph
Let $G$ be an undirected, simple, bipartite graph with parts $V$ (having $n$ vertices) and $W$ (having $m$ vertices). Define the following $n$-by-$n$ matrix: for any $i,j \in V$, $$a_{ij} = |N_i \cap ...
- 1,068
2
votes
1
answer
358
views
positive semidefinite matrix condition
There is a great work of Alizadeh that in section 4 speaks about Minimizing sum of the first few(k-largest) eigenvalues of a symmetric matrix. Instead of a symmetric model we use the weighted ...
- 161
2
votes
0
answers
172
views
Second eigenvalue of a weighted tree
Hello,
I am interested in upper bounding the second largest eigenvalue of the adjacency matrix of a graph $T$ with the following property:
1. $T$ contains self loops.
2. $T$ contains multiple edges (...
- 225
2
votes
0
answers
143
views
Optimization over Spectral Laplacian in cycles and trees
Is there any idea on how one can deal with an optimization problem of sum of k largest eigenvalues(min) of Laplacian matrix of a simple cycle or tree?
I would like to use semidefinite programming for ...
- 161
0
votes
0
answers
146
views
Global solution for spectral clustering
I used spectral clustering for directed graphs suggested by Dengyong Zhou paper to partition the graph.I selected the eigen vectors corresponding to k largest eigen values and then I use kmeans or FCM ...
- 13
3
votes
1
answer
2k
views
Singular values of differences of square matrices
Suppose $A, B \in \mathbb{R}^{n \times n}$. Let $\sigma_1(A),\ldots,\sigma_n(A)$ be the singular values of $A$, and let $\sigma_1(B),\ldots,\sigma_n(B)$ be the singular values of $B$. If I know these ...
- 794
13
votes
2
answers
1k
views
Combinatorial proof of (a special case of) the spectral theorem?
The spectral theorem for a real $n \times n$ symmetric matrix $A$ says that $A$ is diagonalizable with all eigenvalues real. If $A$ happens to have non-negative integer entries, it can be interpreted ...
- 118k
22
votes
2
answers
4k
views
Fast Fourier transform for graph Laplacian?
In the case of a regularly-sampled scalar-valued signal $f$ on the real line, we can construct a discrete linear operator $A$ such that $A(f)$ approximates $\partial^2 f / \partial x^2$. One way to ...
- 3,059