Questions tagged [adjacency-matrices]

An adjacency matrix is a square (0,1)-matrix used to represent a graph. Its entries indicate whether pairs of vertices are adjacent or not.

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Regularize a graph while embedding the spectrum of adjacency matrix

Given an irregular graph $G$ whose maximum degree is $d$, I am interested in producing a new graph $G'$ which is regular and has the spectrum of the adjacency spectrum of $G$ embedded in the spectrum ...
Chaithanya's user avatar
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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 ...
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Eigenvalue decomposition of normalized adjacency matrix

Let $A$ be an adjacency matrix of undirected graph $G$, where $G$ is a connected graph. The normalized adjacency matrix is defined as $\hat{A}=D^{-1/2}AD^{-1/2}$, where $D$ is degree matrix of graph $...
MikeDean's user avatar
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(How) does the spectral gap of the $n\times n$ Rubik's cube close with $n$?

Consider the spectrum of the adjacency matrix $A$ of the Cayley graph of the standard, 3x3x3 Rubik's cube generated with the usual quarter-turn and half-turn twists of each face (the Singmaster ...
Mark S's user avatar
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Show that two matrices are strongly shift equivalent

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 ...
Sanae Kochiya's user avatar
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Solution to the sum of k-step path lengths between node pairs in directed weighted graphs

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 ...
chrisLoweDev's user avatar
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Formulas to determine the value of graph energy with addition or deletion of edges

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 $ ...
gete's user avatar
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Computing adjacency matrix eigenvalues by counting closed walks

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 $...
Sam Hopkins's user avatar
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24 votes
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Factorization of the characteristic polynomial of the adjacency matrix of a graph

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 ...
Joe Silverman's user avatar
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Eigenvalues of the complement of a graph

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 $A$ ...
GA316's user avatar
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Digraphs with unique walk of length $k$ between any two vertices

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 $...
Antoine Labelle's user avatar
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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 ...
H A Helfgott's user avatar
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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
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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 ...
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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 ...
rysuds's user avatar
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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 ...
vidyarthi's user avatar
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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&...
Amin235's user avatar
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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 ...
Gabe K's user avatar
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3 votes
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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 ...
Student88's user avatar
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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 ...
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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 ...
Ludwig's user avatar
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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 ...
Elayeek's user avatar
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8 votes
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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 ...
Delio Mugnolo's user avatar
11 votes
5 answers

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 ...
Mustafa Said's user avatar
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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: ...
Penguian's user avatar
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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 ...
xzhh's user avatar
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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, ...
Delio Mugnolo's user avatar