Hi I need the bound for the spectrum gap of random semiregular ($\ell$, $r$)-bipartite graph. This paper (http://arxiv.org/abs/1212.5216) gives the bound for $\ell$-regular bipar …

Let $W$ be a weighted adjacency/affinity matrix for some connected graph. $W$ is symmetric and non-negative. If $W_{ij}$ is large, then vertex $i$ and vertex $j$ have high affinity …

Consider the hypercube $Q_4$. I would like to know how to compute the number of steps of a random walk in this graph such that the probability to be at a vertex is a given number …

$A$ is a non-negative, integer, irreducible, $m$ by $m$ matrix. It is well known (Perron-Frobenius) that $A$ has a positive eigen value (denote it by $\lambda$) with a positive eig …

Consider an integer cube of size $\sqrt{k} \times \sqrt{k} \times \sqrt{k}$, where $k$ is an asymptotically large perfect square number. Place k points in this cube at uniformly ra …

It is well-known that a random graph a.e. has diameter 2. It is also well-known that the number of distinct eigenvalues of a graph is at least the diameter plus one. But what is k …

I was wondering how to relate the spectra of the composition of two graphs in term of the factors...someone can help me?

In Gowers' paper on quasirandom groups, he suggests a spectral theory of bipartite graphs employ the singular values of the bipartite adjacency matrix. Accordingly, singular value …

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 …

Let $A$ be the adjacency matrix of a strongly regular graph. When is it possible to sign $A$ (i.e. replace some of the +1 entries by -1) so that the resulting matrix has exactly t …

I have two specific queries: When computing spectral density function for undirected graphs, we have \begin{equation} \label{specdensity} \rho(\lambda) = \frac{1}{n}{\sum_{j=0} …

In the study of representation theory of $S_n$, we know that the irreducible characters of $\chi_\lambda$ of $S_n$ are indexed by partitions $\lambda \vdash n$. There are several m …

I have a weighted Laplacian matrix $L$ of a strongly connected directed graph and a diagonal matrix $D$ with positive entries. Since the graph is directed, $L$ is non-symmetric rea …

I have a graph with multiple connected components, and its adjacency matrix. I form the Laplacian matrix (wiki Laplacian matrix), and from the 1K nodes there around 100 eigenvalues …

Let $G = (V,E)$ be a connected simple graph (unweighted, undirected, no selfloops) on $n$ nodes. Let $\mathbf{d} := (d_1, d_2, ..., d_n) \in \mathbb{R}_{>0}^n$ be a vector of arbit …