All Questions
Tagged with matrices graph-theory
102 questions
2
votes
0
answers
35
views
Limiting spectral distribution of a random matrix with specific structure
First, consider an $N \times N$ Hermitian random matrix $V$ from the Gaussian Unitary Ensemble (GUE). It is well known that the empirical spectral distribution of the GUE satisfies the semicircle law ...
- 3
4
votes
1
answer
420
views
Visualizing the elements of a finite group and does the Gram matrix determine the finite group?
Let $G$ be a finite group with $n = |G|$ elements. By Cayley's theorem for finite groups, we have an injective homomorphism of groups:
$$
\pi : G \rightarrow S_n, \quad g \mapsto \pi(g)
$$
where ...
- 5,409
1
vote
0
answers
48
views
How to solve a optimal matching problem where the "quality" of matching is only determined once all nodes are matched
Note: I'm not a mathematician; I'm just a biologist with basic math background trying to solve a scientific problem, so please excuse my ignorance.
The gist of the problem is as follows:
I have two ...
- 5,429
2
votes
2
answers
461
views
On Knot Equivalence problem statement
How is the knot equivalence problem represented?
By this I mean I am looking for an analogy that compares with graph equivalence. For graph equivalence, we have two graphs $G_1$ and $G_2$ with ...
- 2,934
1
vote
0
answers
134
views
Number of ways to place 4 kings on nxn chessboard
I have a $n\times n$ chessboard and 4 kings inside it. My goal is to count the number of arrangements where some of them are non-attacking or mutually attacking, for example:
In the case where the $4$...
- 47
3
votes
2
answers
197
views
Proper Latin sub-squares of generalized Latin squares
Say we have a generalization of a Latin square, where the square is of size $n \times n$, $n = ab$ and each row and each column has $b$ occurrences of each of $[1, \dots, a]$. Is there always ...
- 341
1
vote
0
answers
255
views
Interpreting positive semidefinite matrix as a graph
Given any symmetric matrix $S \in \mathbb{R}^{n \times n}$, if $S \succeq 0$, is there a way to encode $S$ into a graph such that it takes into account the positive semidefinite constraint, and ...
CommunityBot
- 1
0
votes
0
answers
37
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Largest root of the Adjacency matrix of two graphs (comparison)
Let $G$ and $H$ be two graphs whose spectral radius (largest eigenvalue) of the adjacency matrix is the largest root of the following polynomial:
$$P_G(x) = x^6-x^5-(2a-n+5)x^4+(2a-n+1)x^3+2(5a-3n+5)x^...
- 199
10
votes
2
answers
2k
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When are the adjacency matrices of non-isomorphic graphs similar?
From Wikipedia.
In linear algebra, two n-by-n matrices A and B are called similar if
$$ B = P^{-1} A P$$
for some invertible n-by-n matrix $P$.
If $P$ is a permutation matrix, $A$ and $B$ are ...
- 175
28
votes
3
answers
2k
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Is every positive integer the permanent of some 0-1 matrix?
In the course of discussing another MO question we realized that we did not know the answer to a more basic question, namely:
Is it true that for every positive integer $k$ there exists a balanced ...
- 82.6k
1
vote
0
answers
122
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Hard instances for this graph isomorphism algorithm based on powers of weighted adjacency matrices?
In short, I found an algorithm for GI and the only hard instances
I found so far are non-isomorphic strongly regular graphs with
large automorphism groups.
Q1 What are hard instances for the ...
- 25.4k
2
votes
0
answers
203
views
Permutation similarity of matrices with many distinct entries
This is related to graph isomorphism.
Here matrices are square $n \times n$ with non-negative integer entries.
Two matrices $A,B$ are permutation similar if there exist
permutation matrix $P$ such ...
2
votes
1
answer
133
views
Is the sum of the circulant matrix with a super upper triangular matrix diagonalizable?
By the circulant matrix $C$ in $M_n(\mathbb{R})$, we mean that
$$C=[e_n|e_1|\cdots|e_{n-1}]$$ where $e_1,\cdots,e_n$ are the standard basis vectors in $\mathbb{R}^n$. It is well-known that
$$C=\...
- 16.2k
8
votes
3
answers
8k
views
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 ...
11
votes
5
answers
2k
views
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 ...
- 37.7k
6
votes
1
answer
515
views
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 ...
11
votes
1
answer
467
views
Correspondence between matrix multiplication and a graph operation of Lovász
In his book "Large networks and graph limits", Lovász describes a multiplication operation (he calls it concatenation) on "bi-labeled graphs". An $(m,n)$ bi-labeled graph is a ...
- 12.9k
1
vote
1
answer
143
views
Eigenvalues of directed graph with one outward edge for each vertex
I am concerned with unweighted directed graphs where each node contains exactly one edge pointing to another node, which could be itself. In other words, each row of the adjacency matrix contains one ...
- 12.7k
3
votes
1
answer
102
views
matching two positive-semidefinite matrices
Let $M_1$ and $M_2$ be two real positive-semidefinite matrices. Is there any algorithm to compute a permutation matrix $P$ to minimize $\| M_1-PM_2P^T \|_F^2$ or equivalently to maximize $trace(...
- 166
21
votes
5
answers
2k
views
The middle eigenvalues of an undirected graph
Let $ \lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_{2n} $
be the collection of eigenvalues of an adjacency matrix of an undirected graph $G$ on $2n$ vertices. I am looking for any work or references ...
- 4,713
2
votes
1
answer
156
views
Minimal Laplacian spread of a graph
Laplacian spread of a graph is the difference among the largest and the second smallest Laplacian eigenvalue of the graph. Is there any result or conjecture that discusses about the graphs having ...
CommunityBot
- 1
5
votes
1
answer
382
views
Is it possible to compute a valid Laplacian matrix from an effective resistance matrix?
I am wondering whether it is possible to retrieve a node-admittance matrix $G$ (also called Laplacian matrix) in a purely resistive network composed of nets $\{1, \dots, i, \dots, j, \dots, n\}$, from ...
- 2,175
2
votes
1
answer
407
views
A property of directed acyclic graph
Suppose $A \in \mathbb{R}^p$ is the adjacency matrix of a weighted directed acyclic graph $D$ with vertex set $\left\{v_{1}, v_{2}, \ldots, v_{p}\right\}$, i.e.
$$
a_{i j}=\left\{\begin{array}{lr}
...
- 767
2
votes
1
answer
155
views
What is the name of a matrix operation using the OR operator instead of addition?
Let's say we have two matrices $M$ and $G$ with $G, M \in \{0, 1\}^{n, n}$, we denote by $m_{i, j}$ the element of $M$ in the $i^\text{th}$ row and $j^\text{th}$ column, same for $G_{i, j}$.
Let's ...
- 13.2k
4
votes
2
answers
600
views
Co-trees of a simple graph
Consider fundamental cycles (say $k$ of them) of a specific spanning tree of a simple graph (with $m$ edges) which is also connected and has no one-edge bonds.
Make the graph directed (in an arbitrary ...
- 419
2
votes
0
answers
72
views
How to delete the maximum number of rows of a Boolean matrix by maintaining the sum greater than zero in each column
I have a Boolean matrix (entries are "0" or "1"), which is not square. The sum over each row and each column is constant (but they can be two different values). I would like to ...
- 39.9k
3
votes
2
answers
316
views
Relation graph isomorphism to discrete logarithm
$\DeclareMathOperator\ora{ora}$Let $A_0$ be the adjacency matrix of graph $G$ and $P_0$
permutation matrix of multiplicative order $\rho$.
Let $X$ be positive integer and $B_0=P_0^X A_0 P_0^{-X}$.
Q1 ...
- 37.7k
6
votes
1
answer
726
views
Combinatorics and symmetry in matrices under row and column swaps
Suppose we have a $m\times n$ matrix and a sequence of numbers with which to fill the matrix, $\{c_1,c_2 \dots c_k \}$. I like to think of the numbers as colors, hence the notation. How many unique ...
- 24.2k
2
votes
0
answers
116
views
Intuition behind eigenvalues of a graph mattering
Is there a good intuition behind why the eigenvalues of a matrix corresponding to a graph tell us useful information about the graph? There are a lot of results relating the eigenvalues of the ...
- 6,969
1
vote
1
answer
165
views
Combinatorial graph optimization problem on integer adjacency matrices
We are given a $n\times n$ symmetric matrix $M$ whose entries are positive integers.
Let $z_{i,j}:=\frac{M_{i,j}}{M_{i,j}+\sum_{k\neq i,j}\min(M_{i,k},M_{k,j})}$ for all $1\le i<j \le n$, and
$z:=\...
- 7,226
12
votes
2
answers
2k
views
Integral positive definite quadratic forms and graphs
Let me start with a question for which I know the answer. Consider a symmetric integral $n\times n$ matrix $A=(a_{ij})$ such that $a_{ii}=2$, and for $i\ne j$ one has $a_{ij}=0$ or $-1$. One can ...
- 4,713
4
votes
0
answers
59
views
Graph-class defined by matrix-like vertex-operations
Let $m$ be a positive integer. We define a (directed) graph on $m(m-1)$ vertices
$$V = \bigl\{(i,j) \mid i \ne j,\, i,j\in\{1,\dots,m\}\bigr\}$$
and edges as follows:
$(i,j) \in V$ is adjacent (...
- 1,444
2
votes
2
answers
330
views
Polynomial time algorithm for rigid graph isomorphism
We found, implemented and tested algorithm
for graph isomorphism and it appears to be polynomial
time if the graph is rigid.
Q1 Is the algorithm below correct and polynomial time for rigid graphs?
A ...
- 25.4k
1
vote
0
answers
140
views
Adjacency matrix/tensor operations for graph sequences?
Consider a graph $G=(V,E)$. Its adjacency matrix $A$ is defined by $A_{u,v} = 1$ if $(u,v)\in E$, $0$ otherwise.
Consider a vector $x$ that associates a value $x_v$ to each vertex of $G$. Consider the ...
- 1,444
4
votes
2
answers
1k
views
What's the full assumption for Laplacian matrix $L=BB^T=\Delta-A$?
Graph with no-selfloop, no-multi-edges, unweighted.
directed
For directed graph Adjacency matrix is a non-symmetric matrix $A_{in}$ considering indegree or $A_{out}$ considering outdegree. Degree ...
- 101
34
votes
1
answer
789
views
Which graphs on $n$ vertices have the largest determinant?
This is a question that seems like it should have been studied before, but for some reason I cannot find much at all about it, and so I am asking for any pointers / references etc.
The determinant of ...
- 3,065
0
votes
0
answers
425
views
Linear independence of vectors in Graph Theory
I have poste this question on StackExchange but there were no takers - would I be luckier on this site?
Most of this is well known, so let me just restate the corresponding Math:
Given a connected, ...
- 419
0
votes
2
answers
163
views
Graphs vs matrices (when $0$ weight edges are allowed) [closed]
EDIT:
I have asked for closing this question, and posted an improved version on math.se.
I hope this is ok.
It is often claimed, including by myself for the last 20 years, that matrices are ...
- 1,444
2
votes
0
answers
119
views
Complete graph invariant based on integer programming?
Roughly speaking, we are trying to find complete graph invariant
as the lexicographically first matrix from the permutations
of the adjacency matrix.
Let $G$ be graph, possibly directed graph, of ...
- 25.4k
13
votes
0
answers
237
views
A Dynkin type classification result in linear algebra
Let $G$ be a finite directed acyclic graph. The Cartan matrix $C_G=C$ of $G$ is defined as the matrix with rows and colums indexed by the vertices of $G$ and $c_{i,j}$ counts the number of paths from $...
- 26.5k
2
votes
0
answers
154
views
What characterizes the incidence matrix of a tripartite hypergraph?
The incidence matrix of a graph $G = (V,E)$ is a matrix with $|V|$ rows and $|E|$ columns, in which element $v,e$ is $1$ if node $v$ is incident to edge $e$, and $0$ otherwise.
In bipartite graphs, ...
- 4,746
3
votes
0
answers
108
views
Positive vector in the kernel of an skew-symmetric incidence matrix
Let $G=(V,A)$ be an oriented graph, stronlgy connected with $n\in\mathbb{N}^*$ vertices. Let $M\in\mathcal{M}_n(\mathbb{R})=(m_{i,j})$ be an skew-symmetric matrix of size $n$ and rank $r$, such that ...
- 449
1
vote
1
answer
847
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
- 116
4
votes
1
answer
140
views
Counting adjacency matrices
Here is a question that has come up in the context of a problem that involves counting partially ordered sets.
For an adjacency matrix $A$, let $p$ be the sum of elements in the strict upper ...
- 480
2
votes
0
answers
515
views
Normalized Laplacian matrix versus walk Laplacian matrix (or normalized adjacency matrix versus walk adjacency matrix)
In graphs, found that two different normalization matrices exist for Laplacian and adiacency matrix. I will ask about the adjacency matrix (for the Laplacian matrix the questions are the same). The ...
2
votes
2
answers
234
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 ...
- 37.7k
7
votes
0
answers
177
views
Matrix of high rank mod $2$: must it have a large non-singular minor (with disjoint rows and columns)?
Let $A$ be a $2n$-by-$2n$ matrix with entries in
$\mathbb{Z}/2\mathbb{Z}$ such that, for every $2n$-by-$2n$ diagonal
matrix $D$ with entries in $\mathbb{Z}/2\mathbb{Z}$, the matrix $A+D$
has rank $\...
- 20.2k
1
vote
1
answer
60
views
Rank and edges in a combinatorial graph?
Fix a $d\in\mathbb N$ and consider the matrix $M\in\{0,1\}^{2^d\times d}$ of all $0/1$ vectors of length $d$. Consider the matrix $G\in\{0,1\}^{n\times n}$ whose $ij$ the entry is $0$ if inner product ...
- 13.6k
3
votes
1
answer
336
views
Eigenvalues of random graphs
At time $t=0$, let $G_n(V,E)$ be a graph with $n$ vertices and $m < n$ edges. Then there exists a unique symmetric adjacency matrix $A_n$ associated with $G_n(V,E)$, defined as follows: $a_{ij} = 1$...
0
votes
1
answer
154
views
Energy of a symmetric matrix with $0$, $1$ or $-1$ entries
I have a symmetric matrix with entries $0$, $1$ or $-1$ which appeared in my works in graph theory (the diagonal entries are all zero). I need a good upper bound for the energy of this matrix; i.e. "...
- 2,183