# Tagged Questions

**3**

votes

**2**answers

80 views

### eigenvalue estimate of the adjacency matrix

The adjacency matrix of a nonempty (undirected) graph has a strictly positive largest eigenvalue $\lambda_\max$. A very easy upper estimate for it can be obtained directly by Gershgorin's theorem:
$$
...

**3**

votes

**1**answer

109 views

### The spectral radius of a modified graph

Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$.
This situation has many different names: $G$ is called the cone or the ...

**0**

votes

**0**answers

95 views

### How could I prove this equality for eigenvalues of Laplacian matrix?

I would be glad if you have some comments that how I could prove following statement.
Suppose that graph $G =(N, E)$ be given. The the following program computes the $k$-smallest eigenvalues of the ...

**2**

votes

**1**answer

189 views

### Graph of Grassmannian

Let p be an integer, and let G be the graph $(V=Gr(k,\mathbb{F}_q ^n),E)$
where: $Gr(k,\mathbb{F}_q ^n)$ is the set of all subspace of $\mathbb{F}_q$ of dimension k, and $E=\{ W_1,W_2 \in V | ...

**9**

votes

**1**answer

313 views

### Coherence between different ranking methods of a graph's vertices

Given a (connected) graph $G$ it is natural to want to rank its vertices, with the more "central" vertices ranked higher.
Two natural ways of doing it are:
By the degrees.
By the entries in a ...

**0**

votes

**0**answers

66 views

### Constructing a digraph from its spectrum

This is related to the following question from cs theory stack exchange:
http://cstheory.stackexchange.com/questions/3742/reverse-graph-spectra-problem
So it seems as if given a sequence of real ...

**3**

votes

**0**answers

232 views

### A problem on graph theory and complex numbers!

Let ${\mathcal G} = ({\mathcal V},{\mathcal E})$ be a simple connected undirected graph with $n$ vertices. Also let $z_1, \ldots, z_n \in {\mathbb C}$ be complex numbers such that
$$
||z_1||=\ldots = ...

**4**

votes

**1**answer

210 views

### What is the largest possible operator norm of a sparse (0,1)-matrix?

Inspired by this question, I was wondering about the following problem:
Consider all $n\times n$ $(0,1)$-matrices with $k$ ones. Which of these matrices has the largest operator norm? And how does ...

**-1**

votes

**1**answer

217 views

### Upper bound on iterations count for power iteration algorithm

I'm stuck trying to get upper bound on iterations count for power iteration algroithm for finding first eigenvalue of adjacency matrix $A$ given tolerance value. I've tried to figure something out ...

**4**

votes

**0**answers

124 views

### Relaxation = absorption?

Let $A$ be a stochastic matrix, that is, the entries are non-negative and each row adds to $1$. Assume that it is primitive, that is, $A^n$ has only positive entries for sufficiently large $n$. We ...

**4**

votes

**1**answer

512 views

### How many distinct eigenvalues does a random graph have?

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 known about the ...

**0**

votes

**0**answers

90 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 ...

**1**

vote

**2**answers

250 views

### How to identify bridge nodes between nearly connected graph components in partitioned adjacency matrices?

I have adjacency matrices which have nearly connected components. That is partitions with a dense number of edges between nodes in the same group and few edges acting as bridges between these groups. ...

**0**

votes

**1**answer

322 views

### Find edge weights that fit given node weights

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 arbitrary given node ...

**3**

votes

**1**answer

603 views

### Connection between eigenvalues of matrix and its Laplacian.

Hello!
There are two definitions of graph spectrum:
1) Eigenvalues of adjacency matrix $A$.
2) Eigenvalues of Laplacian of adjacency matrix ($L$).
Different sources offer different properties based ...

**1**

vote

**1**answer

140 views

### a variation on the theory of equitable partitions for graphs

Assume you have a graph with an equitable partition with respect to cells $V_1,\ldots,V_n$. Accordingly, you can take the cellwise average value of a function on the node set - in other words, the ...

**3**

votes

**0**answers

110 views

### inverse M-matrix times mixed-sign vector

Recently a colleague and I came across this unusual phenomenon.
Take $M\in\mathbb{R}^{n\times n}$ a singular irreducible M-matrix, and $b\in\mathbb{R}^{n}$ such that the system $Mx=b$ is solvable ...

**1**

vote

**1**answer

573 views

### equitable partitions

It is well known that if $\pi$ is an equitable partition of a graph, then the spectrum of the corresponding partition matrix is a subset of the spectrum of the graph's matrix (where the matrix can be ...

**2**

votes

**2**answers

521 views

### Positive semidefinite decomposition, Laplacian eigenvalues, and the oriented incidence matrix

Suppose $A\in\mathbb{C}^{n\times n}$ is Hermitian and positive semidefinite with some decomposition $A=BB^*$, where $B=(b_{ij})\in\mathbb{C}^{n\times m}$ (not necessarily the Cholesky decomposition). ...

**3**

votes

**2**answers

708 views

### Complex Eigenvalues of Directed Graphs

I have been computing eigenvalues of adjacency matrices for several directed (not necessarily strongly connected) graphs and one remarkable property seemed to hold (each graph that I have examined ...

**4**

votes

**3**answers

611 views

### Spectral radius of a proper subgraph

I came across a Chinese reference in the paper "On the spectral radius of trees with ﬁxed diameter" by Guo and Shao. The attribute the following to Q. Li, K.Q. Feng in: "On the largest eigenvalue of ...