# Tagged Questions

**3**

votes

**1**answer

97 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 ...

**6**

votes

**1**answer

129 views

### A family of skew-symmetric matrices corresponding to cycles in graphs

When investigating loops in Markov chains I ran into the following observation.
A cycle in a graph $G$ with $n$ vertices may be represented by a matrix $\Gamma \in \mathbb R^{n \times n}$ having the ...

**3**

votes

**2**answers

240 views

### Which directed graphs have a normal adjacency matrix?

I am working on a problem in matrix analysis and I am looking for certain types of normal matrices. I suspect that these "special" normal matrices arise as adjacency matrices of certain graphs. My ...

**3**

votes

**0**answers

157 views

### Beating Kadane's Algorithm

I am seeking some reference on already existing work for the following problem.
Given an $n$-dimensional square matrix $A=DP$ where $D$ is a diagonal and $P$ is a permutation matrix (think of Gaussian ...

**4**

votes

**1**answer

187 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

189 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 ...

**3**

votes

**0**answers

411 views

### determinant of fibonacci-sum graphs

We have a simple graph with vertices $\{v_1, v_2, ... v_n\}$.
The adjacency matrix of this graph is $A= (a_{ij})$ so that
$a_{ij}=1$ if $i+j$ belongs to the Fibonacci sequence;
$a_{ij}=0$ ...

**0**

votes

**1**answer

133 views

### two non-degenerate quadratic forms on $GF(2)^2r$

I know this:
There are two non-degenerate quadratic forms on $GF(2)^2r$. The hyperbolic form may be taken to be
$Q^+(x)=x_0 x_1 + \cdots +x_{2r-2}x_{2r-1}$ ,
and the elliptic form to be
...

**5**

votes

**1**answer

328 views

### Repeated Second Eigenvalue of the Adjacency Matrix of a Graph

This question is motivated by a talk I went to earlier today.
Suppose we have a $d$-regular graph $G$ with $n$ vertices, with adjacency matrix $A$.
Let $$\lambda_1\geq \lambda_2 \geq\dots \geq ...

**2**

votes

**0**answers

211 views

### Matrix Operations Preserving Hurwitz Stability

I begin with terminology I use in the question. A real square matrix $A$ is
negative-stable if for every eigenvalue $\lambda$ of $A$, ${\mathrm{Re}}(\lambda) < 0$;
$\ast$-negative-stable ...

**2**

votes

**2**answers

481 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). ...

**2**

votes

**2**answers

221 views

### Ax=0, estimate min(Hamming(x)) ? Equivalently: Bipartite graph. How to find (estimate) minimal number of vertices1 which are connected with EVEN number of vertices2 ? Equivalently: estimate minimal weight of error correcting code ?

Consider system of linear equations Ax=0 over $F_2$ (field with two elements {0,1}).
Where number of variables is bigger than equations - so we have many solutions $x$.
Question How to estimate ...

**6**

votes

**1**answer

277 views

### Maximal disarrangement of $n \times n$ numbers

This question is inspired by
Martin Erickson's
question,
"Labeling a Square Array."
I'll start by quoting Martin:
the $n^2$ cells of an $n \times n$ array are labeled with the integers
$1, ...

**2**

votes

**2**answers

566 views

### Computing the multiplicity of an eigenvalue of a 0-1 symmetric matrix…

When we want to compute the multiplicity of an eigenvalue of a 0-1 symmetric matrix (viewed as the adjacency matrix of an undirected regular graph), we commonly resort to the know lemma of Feit and ...

**4**

votes

**0**answers

406 views

### Smallest matrix covered by many random n by n matrices

We say that a matrix $M$ can be covered by a (smaller) matrix $N$ if every entry in $M$ is contained in some submatrix of $M$ that exactly equals to $N$, up to reordering the rows and columns of $N$. ...

**23**

votes

**4**answers

2k views

### Adjacency matrices of graphs

Motivated by the apparent lack of possible classification of integer matrices up to conjugation (see here) and by a question about possible complete graph invariants (see here), let me ask the ...

**3**

votes

**1**answer

224 views

### Finding a 5-cycle in a sparse graph efficiently.

Hi,
I was reading this thread: Finding a cycle of fixed length
I want to find a 5-cycle in a graph. Actually, what I really want is a shortest odd cycle of length at least 5, but maybe that is a ...

**6**

votes

**0**answers

587 views

### inverse eigenvalue problem on graph laplacian

I am trying to construct a graph Laplacian matrix from a set of eigenvalue. I've read several papers about inverse eigenvalue problems but to be honest I didn't understand clearly. Could somebody ...

**1**

vote

**5**answers

2k views

### Nodes clusters with a distance matrix

Hi,
I have a (symmetric) matrix $M$ that represents the distance between each pair of nodes. For example,
A B C D E F G H I J K L
A 0 20 20 20 40 60 60 60 100 120 ...

**11**

votes

**2**answers

1k views

### Integral positive definite quadratic forms and graphs

Let me start with a question for which I know the answer. Consider a symmetric integral $n\times n$ matrix $A=(a_{ij})$ such that $a_{ii}=2$, and for $i\ne j$ one has $a_{ij}=0$ or $-1$. One can ...

**3**

votes

**0**answers

299 views

### Combinatorics of signed oriented graphs/skew-symmetric matrices

Consider a "complete" signed graph on $n$ vertices indexed by
$1,2,\dots,n$, that is, a graph in which any two distinct vertices $i$ and $j$ are connected by an oriented edge.
For each pair of ...

**1**

vote

**5**answers

1k views

### Properties of Graphs with an eigenvalue of -1 (adjacency matrix)?

I am wondering if there are special classes of graphs that have eigenvalue of -1 for the adjacency matrix. I know that the complete graphs, Kn, have this property, but am wondering if other graphs do ...