Questions tagged [adjacency-matrices]

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2
votes
1answer
104 views

Traces and closed walks that do not close before their time

Let $A$ be the adjacency matrix of a graph. Then, as is well-known and trivial to show, $\mathrm{Tr} A^k$ equals the number of closed walks of length $k$. Is there a similar way to express (a) the ...
1
vote
1answer
156 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
1answer
127 views

Leading eigenvector value problem as an optimisation problem for asymmetric matrices

As noted in 1806.05647, given a symmetric matrix $A$, the leading eigenvector value problem (LEVP) $$Av = \lambda v,$$ where $A = A^T \in \mathbb{R}^{n \times n}$, $\lambda$ is the largest ...
0
votes
1answer
172 views

Significance of the Eigenvalues of the adjacency matrix of a weighted di-graph

I'm currently running a simulation on a bunch of randomly generated points, each with two randomly selected 'partners' from the set of points. In the simulation the points try to move such that they ...
2
votes
2answers
187 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 ...
5
votes
1answer
307 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&...
2
votes
1answer
161 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 ...
3
votes
1answer
555 views

Finding an adjacency matrix whose cube's diagonal is equal to a given vector

How can I find all binary matrices $A$ such that $A^3$ is a non-negative, integer square matrix and $$\mbox{diag}\left(A^3\right)=b$$ for some given vector $b$? Is there a way to characterize all ...
10
votes
1answer
468 views

is it possible to have two non-isomorphic non-regular graphs with the same adjacent spectrum and the same laplacian spectrum?

For two regular graphs $G$ and $H$, it is possible for them to share the same adjacent spectrum and the same laplacian spectrum. While, on the other hand, is it possible to have two non-regular graphs ...
1
vote
0answers
102 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
votes
0answers
136 views

Which functions preserve the connectivity of graphs/components?

I am somewhat stuck working on an issue and would really love some guidance. I will state the problem, my current state and what led to it in case the solution lies beyond where I was looking The ...
0
votes
2answers
140 views

Is there a name for this operation on graphs - e.g. duplication, Kronecker product of graphs?

I am revising a paper where one of the operations performed on a undirected graph with no loops, is to take each vertex, and split it into two vertices, and take each edge and replace it with 4 edges: ...
-3
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1answer
303 views

adjacency matrix of random geometric graphs [closed]

Consider a graph with N nodes. All nodes are distributed as a Poisson point process with density of λ in a L*L area. There is an edge between two nodes if and only if the distance between them is less ...
1
vote
1answer
219 views

power bounded adjacency matrices

A bounded linear operator $T$ on a Banach space $X$ is called power bounded if $\|T^k\|\le M$ for some $M>0$ and all $k\in \mathbb N$. A classical result of Lorch says that if $X$ is reflexive, ...