All Questions
20 questions
6
votes
2
answers
2k
views
How to understand the combinatorial Laplacian $\Delta$ which is defined on the graph?
I have a question about the combinatorial Laplacian $\Delta$ which is defined by
$$\Delta(u,v)=c(u)1_{u=v}-c(u,v)$$
where $u, v$ are some vertices in the graph $G=(V, E)$, and $c(u,v)$ is a ...
- 288
57
votes
4
answers
15k
views
Connectivity of the Erdős–Rényi random graph
It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is ...
- 7,857
2
votes
2
answers
357
views
Is the Erdős–Rényi giant component result applicable here?
Consider a matrix whose elements are independently assigned a value $1$ with probability $p$ and a value $0$ with probability $1-p$.
Define a cluster of cells as a maximal connected component in the ...
- 41
2
votes
1
answer
508
views
Proof and interpretation of the following percolation theory result for $n\times n$ square grid
While I was discussing this question with @JamesMartin, he mentioned a result here that:
In a $n\times n$ finite square grid, if $p\geq p_c+\epsilon$, such
that $\epsilon>0$ and $p_c$ is the ...
user123818
51
votes
3
answers
4k
views
What is the sandpile torsor?
Let G be a finite undirected connected graph. A divisor on G is an element of the free abelian group Div(G) on the vertices of G (or an integer-valued function on the vertices.) Summing over all ...
- 19.2k
25
votes
3
answers
2k
views
Some models for random graphs that I am curious about
G(n,p)
We are familiar with the standard notion of random graphs where you fixed the number n of vertices and choose every edge to belong to the graph with probability 1/2 (or p) independently. This ...
- 24.7k
11
votes
2
answers
714
views
Pursuit-Evasion type game on graph ("Flyswatter game")
An instance of the "flyswatter game" is defined by a graph $G$ and positive integer $k$. There are two players, A (the 'fly') and B (the 'swatter'). Essentially, the fly moves around $G$ and the ...
- 113
7
votes
3
answers
330
views
Quantifying the noninvertibility of a function
Given a function $f$ from a finite set $X$ to itself, it seems natural to consider $\kappa_f := (\sum_{x \in X} |f^{-1}(x)|^2)/|X|$ as a measure of the non-invertibility of $f$: it equals 1 if $f$ is ...
- 19.7k
5
votes
1
answer
209
views
Randomized version of Turán's theorem
Turán's theorem says the following.
Take any natural $n$ and $r$. Suppose that
\begin{equation*}
|G|>\Big(1-\frac1r\Big)\frac{n^2}2, \tag{0}
\end{equation*}
where $|G|$ is the number of edges of ...
- 128k
5
votes
1
answer
980
views
"Nice" eigenvectors for (square of) adjacency matrix of a bipartite graph?
Let $G$ be a bipartite graph, and let $A$ be its adjacency matrix.
I was wondering in this case whether $A^2$ will have nice eigenvectors that reflect combinatorial structure of the graph. I'd be ...
- 53
4
votes
0
answers
1k
views
Number of arrangements that contain at least 1 path from top to bottom of 2D matrix
I have a $n\times n$ matrix of objects. $n'$ objects are black, and the rest $n^2-n'$ are white.
With that information, I can easily calculate the total number of black element arrangements that exist ...
- 47
3
votes
1
answer
206
views
Component properties in Euclidean graphs with distance threshold
In the context of Euclidean graphs with vertices randomly embedded in either a 2D plane (for instance square with length $L$) or in 3D (similarly, cube of side $L$), where an edge between two given ...
- 649
3
votes
1
answer
184
views
Why is number of single cell clusters always greatest in a random matrix?
Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "...
user125648
3
votes
1
answer
153
views
Randomized version of Turán's theorem II
$\newcommand{\om}{\omega}$Let $\om(G)$ denote the number of vertices in a largest clique of an (undirected) graph $G$ with the set $[n]:=\{1,\dots,n\}$ of vertices. Then
\begin{equation}
\om(G)\ge\...
- 128k
3
votes
1
answer
266
views
Probability that a randomly filled Go board has a set of white stones connected through their von Neumann neighborhoods
I have an $N$ by $M$ grid (a Go board for example), where for every square in the grid, I place a white stone with probability $p$ and a black stone with probability $(1-p)$. We call two white stones ...
- 67
2
votes
1
answer
199
views
Average cluster size of a n-size vector
Given a vector of $n$ cells and $k$ elements in it, we can define a cluster of elements as a contiguous sequence of elements inside the vector.
My goal is to calculate the average cluster size for all ...
- 47
2
votes
1
answer
426
views
Random subgraph properties
Consider a graph $G$ of $N$ vertices and $M$ edges, and assume $G$ has typical complex network properties: it is not necessarily connected, but it has a high clustering coefficient and a giant ...
- 129
2
votes
1
answer
90
views
Generalization: (The "number" of) smaller sized clusters in large random binary matrices follow a descending order. Why?
This is a sequel to the question: Why is number of single cell clusters always greatest in a random matrix?
In their answer, @Aaron Meyerowitz came up with a nice strategy to prove why the number of ...
user125648
2
votes
1
answer
299
views
Can this particular random matrix model be converted/related to any existing graph theory model?
Context:
This a sequel to the question: Is the Erdős–Rényi giant component result applicable here?
Consider a matrix whose elements are independently assigned a value
$1$ with probability $p$ ...
user123818
1
vote
1
answer
183
views
Expectation of edge weights on the complete graph
Let $n,k \geq 3$ be positive integers with $n$ much larger than $k$ and consider a random assignment of weights to the edges of the complete graph $K_n$. On each vertex of $K_n$ we attach a random ...
- 26.9k