# Questions tagged [adjacency-matrices]

The adjacency-matrices tag has no usage guidance.

The adjacency-matrices tag has no usage guidance.

23
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Background: Suppose we have a system in which an electron can hop locally on this lattice. Here locally means that the electron can hop up to a short distance. We can make an energy spectrum that ...

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The following question is from Introduction to dynamical systems, written by Michael Brin and Garrett Stuclk.
Given two non-negative integer square matrices $A, B$, we say $A, B$ are elementarily ...

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The Rubik's cube can be generated with quarter-turn clockwise/counterclockwise twists of each of the front, back, left, right, up, and down faces:
$$\langle F_1,B_1,L_1,R_1,U_1,D_1,F_3,B_3,L_3,R_3,U_3,...

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In a non weighted graph, the adjacency matrix ($A$) raised to the power $k$ will return the number of k-step paths between nodes $i$ and $j$ at the entry $a_{ij}$. Is there an equivalent for weighted ...

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If $G$ is a graph, then the graph energy of $G$ denoted by $E(G)$ is defined as the sum of absolute values of eigenvalues of the adjacency matrix of $G$. It is known that $E(G)\geq E(G-v)$, where $ ...

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Let $G$ be a finite undirected graph. A closed walk in $G$ is a walk from any vertex of $G$ to itself. It is relatively straightforward to show that the total number of closed walks of length $k$ in $...

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Let $G$ be a regular graph of valence $d$ with finitely many vertices, let $A_G$ be its adjacency matrix, and let $$P_G(X)=\det(X-A_G)\in\mathbb{Z}[X]$$ be the adjacency polynomial of $G$, i.e., the ...

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Let $A$ and $\widetilde A$ be the adjacency matrices of a graph $G$ and of its complement, respectively.
Is there any relation between the eigenvalues of $A + \widetilde A$ and the eigenvalues of $A$ ...

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Let $G$ be a digraph such that there is an unique directed walk of length $k$ between any two vertices.
Equivalently, if $A$ is the adjacency matrix of $G$, then $A^k$ is the matrix with all entries $...

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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 ...

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If so, could you provide examples and specify the conditions under which this occurs? Thank you in advance

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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 ...

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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 ...

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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 ...

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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
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1
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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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: ...

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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 ...

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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, ...