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 ...
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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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### 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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### 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 ...
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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: ...
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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 ...
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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, ...