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10 questions
2
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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 $G_{i+1}$.
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- 63.9k
5
votes
1
answer
1k
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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 ...
- 101
3
votes
1
answer
271
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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 $G$ ...
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1
vote
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51
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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 ...
- 63.9k
5
votes
1
answer
1k
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The spectrum of the discrete Laplacian
Consider a connected (we define connected components by defining the set of vertices where every vertex has one neighbour) sublattice $V$ of the square lattice $V \subset\mathbb{Z}^2.$
On this we ...
- 24.2k
4
votes
0
answers
126
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An inequality from the "Interlacing-1" paper
This question is in reference to this paper, http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf (or its arxiv version, http://arxiv.org/abs/1304.4132)
For the argument to ...
2
votes
1
answer
1k
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About distinct eigenvalues of a graph
if a graph with adjacency matrix $A$ and Laplacian $L$ has $k$ distinct eigenvalues then does this fact naturally help define or prove existence of a polynomial $p$ of degree $k-1$ such that $[p(A)]_{...
- 441
0
votes
1
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218
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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\,\ ...
- 7,500
2
votes
1
answer
238
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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 ...
- 12.1k
0
votes
0
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146
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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 ...
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