All Questions
8 questions
6
votes
1
answer
515
views
Non-diagonalizability of the adjacency matrix of a directed graph
Let $G$ be a directed graph with no multiple edges or loops and let $P_i$ be its vertices. Let $A$ be the corresponding adjacency matrix of $G$, i.e. $a_{i,j}=1$ if and only if there is a directed ...
- 4,058
1
vote
1
answer
847
views
Do there exist graphs whose adjacency matrix is positive semi-definite? [closed]
If so, could you provide examples and specify the conditions under which this occurs? Thank you in advance
2
votes
2
answers
234
views
Adjacency matrix of total graph
Is there a nice way of relating the adjacency, incidence , Laplacian matrices and other matrices associated to a graph of a total graph with its original graph, or, say, at least relating that of the ...
- 2,089
5
votes
1
answer
425
views
Determining the primitive order of a binary matrix
Let ${\bf A}_n$ be an $2n \times 2n$ matrix that is defined as follows
$$
{\bf A}_n=\left(
\begin{array}{c}
0&0&\cdots&0&0&0&0&1&1\\
0&0&\cdots&0&0&...
- 313
2
votes
1
answer
316
views
When does a row standardized adjacency matrix have a real spectrum?
A colleague in spatial statistics was looking at a map with about 600 regions. For the application she's considering, the induced adjacency matrix had some undesirable properties (where two regions ...
- 6,001
1
vote
0
answers
105
views
Primitivity of $AA^\top$
Let $A\in\mathbb{R}^{n\times n}$ be a non-negative and irreducible matrix. Consider $B:=AA^\top$. It can be proved (I can post a proof if needed) that the following condition is necessary and ...
- 2,712
8
votes
3
answers
8k
views
Spectrum of an adjacency matrix
The adjacency matrix of a non-oriented connected graph is symmetric, hence its spectrum is real.
If the graph is bipartite, then the spectrum of its adjacency matrix is symmetric about 0. A few ...
- 3,388
11
votes
5
answers
2k
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 ...
- 3,699