0
votes
0answers
9 views

Markov chains: identifying a nonregular transitional matrix

I am currently TA'ing for a course in which the students are soon to learn about Markov chains and stochastic matrices. During the sections, it refers to the possible existence of a stable state and ...
0
votes
1answer
19 views

Is the derivative of the characteristic polynomial equal to the sum of characteristic polynomial of principle submatrices?

Let $A$ by an $n \times n$ matrix over the complex numbers and let $\phi(A,x) = \det(xI-A)$ be the characteristic polynomial of $A$. Let $B_i$ be the principal submatrix of $A$ formed by deleting the ...
1
vote
2answers
37 views

Some Matrix product $A \odot B$

I'm confronted with the following problem: Let $G=(V,E)$ be a directed graph with edge costs $c:E\rightarrow \mathbb{R}$ (Negative cycles do not matter). Let $V=\{v_1,\dots,v_n\}$. For Matrices $A$ ...
2
votes
1answer
47 views

about the power of a matrix

Assume that matrix $A$ contains only 0 or 1 elements. Could anyone give me some condition, under which the matrices $A^i$ (for $i=1,2,3,...,k$) still contains only 0 or 1 elements. For example, I ...
1
vote
0answers
31 views

Fast Cholesky Factrorization for Tree Laplacians

Suppose $T_1$ and $T_2$ represent two Laplacian matrices of two spanning trees of $n$ vertices. Since the Cholesky factorization needs $O(n)$ time for each $T_i\ (i=1,2)$ due to the tree structure, ...
0
votes
1answer
9 views

Support of vector $w$ in graph sparsity

I'm reading about graph sparsity and I have one problem in a paper I'm reading I don't understand, maybe someone can clarify: Graph Sparsity: In graph sparsity, we have a directed acyclic graph ...
0
votes
0answers
30 views

A question about similarity transformation.

Say $A$ is an $n\times n$ symmetric matrix such that every row (and hence column) has exactly $d<n$ non-zero entries. Does there exist similarity transformations on $A$ which will maintain these ...
1
vote
0answers
15 views

Prove one of the eigenvector entries has the smallest magnitude

Let $L\in \mathbb{R}^{n \times n}$ be the Laplacian matrix of a simple undirected graph and $D_i$ be the same size matrix with $i$th diagonal element $1$. Denote the smallest eigenvalue of $L+D_i$ as ...
1
vote
1answer
37 views

Kronecker product and the vec operator: confusion on proof of vec(AXB) = (transpose(B) ⊗ A) vec(X)

I was reading up on Kronecker products and vec operator from couple of sources and landed on the equation: vec(AXB) = (transpose(B) ⊗ A) vec(X) suppose ...
1
vote
1answer
67 views

Determine cycle from adjacency matrix

Is there a way/algorithm to determine if there is a cycle in a graph if I only have the adjacency matrix and can not visualize the graph?
0
votes
0answers
28 views

Tightest upper bound for $\sum_j g_{ij}$ of an adjacency matrix of a graph

If I have an adjacency matrix of a graph $G$ (i.e. $g_{ij}=1$ if $i$ and $j$ are connected and $g_{ij}=0$ if not. $g_{ii}=0$), is there any tighter upper bound on $\sum_{j} g_{ij}$ than just $n-1$ ...
1
vote
2answers
69 views

Measuring the entropy of a graph representing a transition probability matrix of a first order markov chain

There's a research project i'm currently working on which requires me to analyze various aspects of "worlds" represented by transition probability matrices, where the nodes represent objects in the ...
1
vote
0answers
60 views

Sparse matrix algorithms involving data-driven or random access / walk

I am looking for some well-known algorithms in which sparse matrix elements are accessed in a non-structured way, i.e. row/column depends on a value of another (sparse) matrix/vector element or some ...
1
vote
1answer
146 views

Tree Traversal - Simple Puzzle type Issue.

This is a puzzle like question,based on Fibonacci like structure of the tree. Actually it is a short question with out any complex concepts. It appears bit big,since I have added explanations with ...
2
votes
0answers
31 views

Counting symmetric binary matrices with constant line-sum

I'm trying to count, as the title suggests, symmetric matrices with entries $0$ or $1$ and constant line-sum $k$. ($0 \leq k \leq n$). If you start listing the number of these on a table you'll get a ...
1
vote
1answer
28 views

Irreducible matrix equivalent connectedness of matrix graph?

If a matrix is irreducible, based on the following definition A matrix is reducible if there are two disjoint sets of indexes $I,J$ with $|I|=\mu$, $|J|=\nu$, $\mu+\nu=n$ such that for every ...
2
votes
1answer
82 views

Checking connectivity of adjacency matrix

What do you think is the most efficient algorithm for checking whether a graph represented by an adjacency matrix is connected? In my case I'm also given the weights of each edge. There is another ...
0
votes
1answer
41 views

Checking the correctness of the adjacency matrix for the given graph

I found the adjacency matrix for this graph; it is shown next to it. Is it correct?
0
votes
1answer
29 views

Constructing an eigenvector for a certain matrix representing a graph with a perfect code

Let $A$ be a symmetric $(0,1)$-matrix whose row sum is $r.$ Suppose I have a $(0,1)$ vector $v$ such that $$Av = \vec{1} - v.$$ By taking the vector $$u = \vec{1} - (r+1)v$$ we see that $$Au = A ...
0
votes
0answers
13 views

compare magnitude of elements of Perron-Frobenious vector

Consider a nonnegative, primitive matrix $A=(a_{ij})_{n\times n}$ with positive diagonals. From the Perron-Frobenious theorem, the spectral radius $\rho(A)$ is an eigenvalue of $A$ and we have a ...
4
votes
1answer
42 views

What is the Laplacian Matrix used for?

You can turn graphs into several matrix forms depending on what data you want to focus on. Does the Laplacian form have any uses on its own, or does it need to be paired with other things as some ...
0
votes
0answers
67 views

Calculating Eigenvector Centrality & Betweenness Centrality formulas explained in simple terms

I'm currently working on a software application that has a function that analyses networks of people and the relationships between them. Two of the important variables we look at are Eigenvector ...
0
votes
0answers
17 views

What is the difference between closed and open subsets in reducibility of a graph

I've read somewhere the following sentence, the graph A is reducible to at two closed subsets. Is that different than just saying "A is reducible" ?? What is the difference between open and closed set ...
1
vote
0answers
52 views

Formula for position in an upper triangular matrix

I'm currently working on the Travelling Salesman's Problem in a computer science module. I have implemented some linear programming techniques using the software lp_solve. I've ended up with an upper ...
0
votes
2answers
56 views

Proving that irreducibility of a matrix implies strong connectedness of the graph [duplicate]

I have tried to prove that if a matrix $A\in\mathbb{C}^{n\times n}$ is such that there are no two sets $I,J\subseteq\{1,\dots,n\}$ that are disjoint, complementary, nonempty, and such that for all ...
0
votes
0answers
29 views

The lower bound of Cheeger Inequality as the degree goes to infinity

Consider an undirected graph $G(V,E)$ with adjacency matrix $A$ and define the graph Laplacian as \begin{equation} L = D - A \end{equation} where $D$ is a diagonal matrix such that $D(i,i) = d_i$. ...
2
votes
2answers
248 views

Matrix graph and irreducibility

How do I prove that if $A\in\mathbb C^{n\times n}$ is a matrix then it is irreducible if and only if its associated graph (defined as at Graph of a matrix) is strongly connected? Update: Seeing as ...
2
votes
1answer
152 views

Graph of a matrix

How to define the graph of a square matrix $\mathbf{G}$ with real entries? I know that given a graph $\Gamma(V, E)$, one can define its adjacency matrix $\mathbf{A}$. But given a matrix $\mathbf{G}$ ...
0
votes
0answers
18 views

Matrix Representations of Chordal Graphs and Uses in Linear Algebra

Chordal graphs have the property of perfect elimination ordering. In Knuth's 2012 Christmas lecture ~1:12:10 he mentions that when the coefficients of a linear algebra problem can be written as a ...
0
votes
0answers
33 views

Eigen vectors of graph laplacians

I have been reading about spectral graph theory from Daniel A. Spielman's notes. Fiedler’s Nodal Domain Theorem from this note says that : Let $G = (V, E, w)$ be a weighted connected graph, and let ...
1
vote
1answer
52 views

$M_{R^n}$; how to derive $n$ for transitive closure?

When finding the transitive closure of a relation $R$, I convert $R$ into a boolean matrix $M_R$, and find the union between $M_R$ and its powers up to $n$. $$M_{R^*} = M_{R^1} \lor M_{R^2} \lor ...
0
votes
1answer
64 views

Calculating Adjacency Matrix

I'm having trouble understanding the concept, I know it is pretty simple but could someone help me out. Assume that I have the following: $V = \begin{bmatrix} 0&0&1 \\ 0&0&1 \\ ...
0
votes
1answer
212 views

Shortest distance matrix given an adjacency matrix?

If I have an adjacency matrix, how can I find a matrix that has the shortest distance between each pair of nodes? (distance matrix, but the nodes are not in a euclidean space) I'm trying to implement ...
1
vote
2answers
74 views

How do I find the adjacency matrix for the nodes of an n-dimensional finite grid?

I have an orthotopic grid, in n-dimensions (usually small ~<3), where each node is connected to it's orthogonal neighbours. The grid may be any number of nodes long, but is finite (and usually ...
1
vote
1answer
85 views

On Adjacency Matrix of a Graph with a Cut Vertex and a Bridge

Let $G$ be a graph. If $v_i$ (resp. $v_iv_j$) is a cut vertex (resp. a bridge) of $G$, what can you say about its adjacency matrix $A(G)$?
1
vote
1answer
47 views

Rank-one modification of graph Laplacian

Suppose I have a Laplacain matrix for a 3-node-path graph as follows $L=\left[\begin{array}{ccc} 1 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 1 \end{array}\right]$ Now, I want to ...
0
votes
1answer
24 views

How to describe symmetric nodes in a graph

For instance, in the path graph $P_4$, node $1$ and $4$ are symmetric, how to mathematically describe this in graph theory? And any algebraic properties related to this? Thanks!
1
vote
1answer
91 views

Permutation matrix and simple directed graph

I have some code that works with simple directed graphs, but it is kinda slow. So I converted it to use an adjacency matrix instead of keeping a list of pairs of nodes. The code finds the ...
1
vote
0answers
61 views

Prove that the minimum of row sums of a nonnegative symmetric matrix is preserved

Let $A$ be an $n\times n$ adjacency (nonnegative, irreducible and symmetric) matrix with zeros on the diagonal. Denote $i$-th row sum of $A^k$ as $r^{(k)}_i$, where $k\geq1$. I want to prove that if ...
1
vote
0answers
113 views

Adjacency matrix of directed graph

I am given adjacency matrix $A$ of directed graph. $A(x,y)$ counts the number of edges from $x$ to $y$. I want to show that if $A$ has constant outdegree $d$: (i) For any eigenvalue $\lambda$, we ...
1
vote
1answer
162 views

Principal EigenVector of an Adjacency matrix of an undirected graph

For an undirected graph, since the adjacency matrix will be symmetric, can we draw any relations between the principal eigenvector and the degree of nodes in the graph. Also can we do the same with ...
0
votes
1answer
128 views

How can one actually use Adjacency Matrix for understanding a graph?

I don't see any real reason why we would use an AM to represent a graph, beside visual appeal and ease. Generally, we would perform matrix operations on Matrices like |A|, Transpose and loads of other ...
1
vote
0answers
144 views

Area optimization: Packing rectangles inside rectangle

Background: I am scrambling to figure out the optimization algorithm to build sprite image, which is essentially a big container rectangular image, with multiple rectangular images. I have found an ...
2
votes
1answer
188 views

Finding characteristic polynomial of adjacency matrix

Short question im having a tad difficulty with. I'm trying to find the characteristic polynomial of a graph that is just a circle with n vertices and n edges. I think the adjacency matrix should ...
1
vote
1answer
81 views

Gauss Elimination for Colorability Problem

Consider the following system of linear equations modulo 2: $A.X + B.Y = Z, $ where $A$ is a non-singular(modulo 2) $n$ x $n$ boolean matrix, $B $ is $n$ x $m$ boolean matrix, $X$ is n-dimensional ...
1
vote
1answer
166 views

Uniqueness of doubly stochastic matrix descomposition

this is my first question in the site. Thanks in advance for all answers. It is well known that each bistochastic matrix can be represented as a convex combination of permutation matrices. I am ...
1
vote
1answer
114 views

Adjacency Matrices

Can someone explain adjacency matrix's in simple terms? I'm not grasping the material from the text at all, and can't solve the sample solutions provided.such as k2,k3 and the reverse. I understand ...
10
votes
1answer
222 views

From matrices to bipartite graphs

Assume $G(A,B)$ is a bipartite graph and assume $L(G)$ is the adjacency matrix of its line graph. define $$B=[3\text{I}+L(G)]^{-1}$$. Is it always the case that for each edge $e=(a,b)\in G$, we have: ...
5
votes
2answers
75 views

Do I influence myself more than my neighbors?

Consider relations between people is defined by a weighted symmetric undirected graph $W$, and $w_{ij}$ shows amount of weight $i$ has for $j$. Assume all weights are non-negative and less than $1$ ...
1
vote
1answer
59 views

Solving a Gaussian elimination problem.

I have been given a graph with n nodes. Now, I have to color every node of this graph by k colors, number from 0 to k-1. Now, there is a rule. For a node $x$ with adjacent nodes $y_1 , y_2, y_3, ...