# Tagged Questions

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### Delocalization of eigenvectors in Expanding Graphs

Given an adjacency matrix A, can we say something about whether the eigenvectors corresponding to its highest(or second-highest) eigenvalues are delocalized ? By delocalization I mean that every ...
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### finding the decomposition of Laplacian matrix with position of zero elements unchanged

I'd like to know whether it's possible to find the decomposition of a Lapalacian matrix $A$ $B^TB = A$ where $B$ has the same dimension with $A$ and the position of zero elements in $B$ is the same ...
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I have a question related to Lemma 4.2 from this pdf (which is, btw quite a nice exposition of Hoffman Singleton work on the classifications of Moore graphs of diameter 2 and 3.) We are given a $n ... 1answer 74 views ### Possible relation between spectra bounds of two matrices A Laplacian matrix$L\in\mathbb{R}^{n\times n}$, is a symmetric matrix with entries, l_{ij}=\begin{cases} 1=\sum_{i,~ i\neq j} w_{ij} &\mbox{if } i=j \\ -w_{ij} & ... 1answer 108 views ### Eigenvalues of a special block matrix associated with strongly connected graph Definition Let$G=(V,E,A)$be a strongly connected directed graph, where$V=\{1,2,...,n\}$denotes the vertex set,$E$is the edge set, and$A$is the associated adjacent matrix with$0-1$weighting, ... 3answers 663 views ### Significance of eigenvalue When I represent a graph with a matrix and calculate its eigenvalues what does it signify? I mean, what will spectral analysis of a graph tell me? 5answers 874 views ### Spectrum of adjacency matrix of complete graph Fooling around in matlab, I did an eigenvalue decomposition of the adjacency matrix of$K_5$. ... 0answers 132 views ### Eigenvalues of regular graphs Could someone give me a hint for exercise 2.iii of these lecture notes? The exercise asks to show that a$k$-regular undirected graph (without loops) whose adjacency matrix$A$has eigenvalues ... 1answer 183 views ### Spielman's proof of graph connectivity I use Spielman's lectures on course Spectral Graph Theory I have few question regarding Lecture 2. The Laplacian, especially Lemma 2.3.1 (Graph connectivity). Please, help me to make it a little bit ... 1answer 134 views ### Two formulas for the minimal eigenvalue of a graph Hello again everybody, I'm reading Fan Chung's monograph Spectral Graph Theory. In it, she has two formulas for the minimal eigenvalue of a graph. She doesn't explain why they're equivalent, and I'm ... 1answer 226 views ### What does the minimal eigenvalue of a graph say about the graph's connectivity? I'm reading Fan Chung's Spectral Graph Theory, and I'm now in chapter 2. There, Chung proves Cheeger's inequality, which is that$2h_G \geq \lambda_1 > h_G^2/2$for any graph$G$. To me, this ... 2answers 173 views ### Graphs with eigenvalues of large multiplicity For a strongly regular graph, there are exactly 3 eigenvalues, all nonzero (I believe). One has multiplicity 1, which means the other two have pretty high multiplicities. There are tables that give ... 1answer 122 views ### Kinks in the eigenvalue spectrum of short range lattices Take a periodic one-dimensional lattice of size$N$with$2k$nearest neighborers. That is, vertex$i$is connected to$i+1,i+2,...,i+k$and$i-1,i-2,...i-k$(with the understanding that the indices ... 1answer 227 views ### eigen decomposition of an interesting matrix (general case) Lets define:$U=\left \{ u_j\right \} , 1 \leq j\leq N= b^{L},$the set of all different sequences of length$L$where each element of the sequence can be an integer in$\left \{ 0, 1, .., b-1 ...
Lets define: $U=\left \{ u_j\right \} , 1 \leq j\leq N= 2^{L},$ the set of all different binary sequences of length $L$. $V=\left \{ v_i\right \} , 1 \leq i\leq M=\binom{L}{k}2^{k},$ the set of ...