# Tagged Questions

**5**

votes

**4**answers

233 views

### Do there exist sparse graphs with large crossing number?

Does there exist a sequence of graphs $\{ G_n \}$ such that
$G_n$ has $n$ vertices,
the number of edges of $G_n$ is $O(n)$, and
the crossing number of $G_n$ is $\Omega(n)$?
In particular, do ...

**4**

votes

**1**answer

111 views

### Expectation of ratio of functions of i.i.d. Bernoullis: a concentration question

Consider the following $n \times n$ symmetric matrix of i.i.d. Bernoulli random variables, $X_{ij}$. For $i=1,...,n$ and $i<j\le n$. Let $X_{ij} \sim \text{Bernoulli}(p)$ when $i \ne j$ ($p$ ...

**2**

votes

**0**answers

27 views

### “drift” of a random graph $G(n,p)$ with $p=\alpha\ln{n}/n$

Suppose $G\sim G(n,p)$ with $p=\alpha\ln{n}/n$ for some large constant $\alpha$. I wish to show a certain "drift" property of $G$, which can roughly be phrased as follows: if $u$ is "not that far" ...

**3**

votes

**1**answer

88 views

### Independent Sets in random geometric graphs

I was wondering if a lot is known about independent sets in Random Geometric graphs? Most google searches don't bring up much. In addition I'm interested in any algorithms used for finding independent ...

**2**

votes

**1**answer

59 views

### Hamiltonicity of random graphs with high girth

We say that $G\sim G_{n,f}$ (for $f=f(n)$) if $G$ is chosen uniformly at random from all graphs on $n$ vertices with girth $g(G)\ge f(n)$. Is there any threshold function $F(n)$ such that when $f\ll ...

**3**

votes

**0**answers

86 views

### Node covering in a random graph

Given $N$ nodes randomly placed in a $D\times D$ area, i.e., the position of each node is randomly chosen. Assume that both $N$ and $D$ are sufficiantly large.
An agent can move in the area at ...

**0**

votes

**2**answers

81 views

### different way of selecting a random graph

Consider having a 'base' graph $G=(V,E)$ and selecting each vertex with independent probability $p$ and having the induced subgraph of $G$ with all 'selected' points as your random graph. Has this ...

**4**

votes

**1**answer

128 views

### Expected number of connected components in a random graph

For a random graph G(n,p) what is the expected number of connected components? What is the probability distribution of this value?
I'm specially interested in what happens for small values of p, ...

**1**

vote

**0**answers

50 views

### Simulation of disassortative random graphs

Recently I have been trying to find a succinct algorithm for generation of disassortative networks. The best I have found is the algorithm by Newman described in his paper "Mixing patterns in ...

**3**

votes

**0**answers

168 views

### Hitting edges in graphs at random and let them die with honor

Let $G=(V,E)$ be a finite simple 2-connected graph. Let $B\subseteq E$ be a set of bad edges of size $k:=|B|$. For each edge $e\in E$ we toss a fair coin ($p=1/2$) and if the outcome is head, we hit ...

**5**

votes

**6**answers

413 views

### Random planar, bipartite graphs

I have a need to generate random planar graphs none of which have an odd cycle,
i.e., bipartite graphs.
I know there is a substantial two-decade literature on random planar graphs, little with which I ...

**3**

votes

**1**answer

226 views

### Expected Value for a Connected Graph

Consider a connected graph of N nodes.
Assign randomly to each node a distinct number from 1 to N.
For each node consider the maximum adjacent value or itself if all adjacent values are smaller.
...

**3**

votes

**3**answers

356 views

### random networks and prime numbers

I have been studying networks recently and accidentally came up with an heuristic approach towards the distribution of prime numbers. The prime number theorem states that
$\pi(n)\sim n/log(n)$ for ...

**4**

votes

**3**answers

169 views

### Tracking automorphism groups of graph processes

Start with an edgeless graph on $n$ labeled vertices, and note that the automorphism group is $\Sigma_n$, the symmetric group on $n$ elements. Now imagine that we randomly start throwing in all of the ...

**0**

votes

**0**answers

59 views

### Maximum Independent set of sparse graphs with few triangles

Notations used
$\alpha(G) = $ Max sized independent set of graph $G$.
$n(G) = $ Number of vertex in graph $G$.
Theorem (by Ajtai et al.): For a triangle-free graph $G$ and max degree being ...

**3**

votes

**0**answers

126 views

### Concentration inequality for function of independent Bernoulli r.v.'s (related to random graph)

Consider a random undirected graph on a set of $n$ nodes, say $\{1,2,\ldots,n\}$, such that the probability of edge between nodes $i$ and $j$ is $p_{ij}$ (we may assume $p_{ij}=o(1)$ for all $i,j$, ...

**1**

vote

**1**answer

457 views

### Probability of two vertices being connected in a random graph

Consider a random directed graph with $N$ vertices where each vertex $v$ has exactly one link to some vertex (maybe to itself) $u$ with known probability $a_{vu}$. What is the probability of ...

**-3**

votes

**1**answer

168 views

### 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 ...

**2**

votes

**0**answers

89 views

### Reachability in dynamic random graphs

There has been some advice on how to ask questions. So I reask my question.I have a problem related to graph theory, random graphs or random geometric graphs in special that really confuses me. ...

**2**

votes

**2**answers

238 views

### Random graphs require O(n log(n)) edges until they are almost certainly fully connected - what are more concrete boundaries ?

As far as I understand my literature, the probability of a random graph being fully connected tends toward one as the number of edges approaches a value of size $O(n \log(n))$
The way I read this, ...

**1**

vote

**0**answers

60 views

### Can this application of the moment closure method to epidemics on networks be made rigorous?

Short version: Consider the SI model of infectious disease spread on a random graph $G$ with a given degree sequence. Let $j$ be a vertex and let $i$ and $k$ be two of its neighbors. If $G$ ...

**2**

votes

**1**answer

250 views

### Removing edges from Erdős–Rényi graph to make two nodes disconnected

Consider a Erdős–Rényi graph on $n$ nodes, say $1,2,\ldots,n$, ($n\geq 3$) such that the probability of edge between any two nodes is $c/n$. I wish to know if there is a result that says
"There ...

**4**

votes

**1**answer

512 views

### How many distinct eigenvalues does a random graph have?

It is well-known that a random graph a.e. has diameter 2. It is also well-known that the number of distinct eigenvalues of a graph is at least the diameter plus one.
But what is known about the ...

**5**

votes

**1**answer

182 views

### The structure of small components in random graphs with a given degree sequence

Background and definitions
Consider a random graph on $n$ vertices with a nicely behaved degree sequence. That is, letting $d_i(n)$ denote the number of vertices of degree $i$, suppose that for all ...

**3**

votes

**0**answers

65 views

### Is there precedent in the literature for a variant of a random geometric graph where vertices are the centroids of discs placed by random sequential adsorption (RSA)?

Imagine I form a random graph by simulating random sequential adsorption (RSA) of discs (each with the same radius $r$) on $[0,1]^2$ until I cover the plane at a density $\leq U$, where $U \leq ...

**0**

votes

**1**answer

106 views

### Is there a proper way to define a threshold vertex density for a random graph s.t. the graph is fully connected?

Imagine one generates some form of random graph (e.g. a random geometric graph) and via simulation, calculates the probability that there exists an edge-wise path between all vertices in the graph as ...

**2**

votes

**2**answers

258 views

### The probability distribution for vertex degree in a unit disc graph generated from random points on a plane

Imagine I cover an arbitrarily large plane with randomly placed points at some density $\rho$ s.t. the number of points in any randomly sampled area $A$ (of arbitrary shape and size) is $\approx ...

**0**

votes

**1**answer

250 views

### Two different definitions of Erdos-Rényi random graph

There are two or more ways to define an Erdos-Rényi random graph. Let consider the following two:
1) $G_n=(V_n,E_n)$ with vertex set $V_n=(1,\dots,n)$ and edge set $E_n=(ij\in\mathcal{P}_2(V_n)\ |\ ...

**0**

votes

**0**answers

191 views

### counting k-cliques not also (k+1) on random graphs

consider the set of graphs with $n$ vertices and exactly half of all $\binom n 2$ possible edges.
looking for a formula that counts the number of these graphs that have a $k$-clique but not a ...

**3**

votes

**1**answer

330 views

### How random are random spanning trees?

Suppose you take a $G(n,p)$ random graph for a fixed probability $p$ and find a spanning tree using Kruskal's algorithm. If you now repeat this process indefinitely, will every tree on $n$ vertices ...

**0**

votes

**1**answer

123 views

### Generating spatially-aware degree-preserving random graphs?

In the study of biological neural networks, researchers sometimes compare hypotheses vs. a degree-preserving random null model. One major criticism against this approach is that connections in neural ...

**4**

votes

**1**answer

255 views

### Cut-distance between two Erdos-Renyi random graphs

Consider two Erdos-Renyi random graphs $G_1,G_2$ on $n$ nodes, with the edges in each graph generated independently at random with probability $1/2$. My question is about the cut-distance between ...

**4**

votes

**0**answers

219 views

### Expected number of components with multiple cycles in a subgraph of a square lattice

Short version
Is there an understanding of the emergence and subsequent disappearance of components with zero, one, or more cycles in a random subgraph of a square or cubic lattice, as the ...

**0**

votes

**0**answers

136 views

### Graph theory meta-question

If property (P) holds for perfect graphs and almost all graphs does it hold for all graphs?

**2**

votes

**1**answer

162 views

### matchings in hypergraphs

I have been reading Pippenger and Spencer's paper "Asymptotic behavior of the chromatic index for hypergraphs" and they comment that their result is applicable to the family of random k-uniform ...

**6**

votes

**1**answer

344 views

### How does the distribution of Erdős number evolve over time ? How to build a model to fit the real data ?

Let $E(n,t)$ be the number of mathematicians with finite positive Erdős number $n$ at time $t$. As old mathematicians leave, and new mathematicians come, how does $E(n,t)$ change over time ?
We can ...

**8**

votes

**1**answer

603 views

### Correlation-Function for Random Graph Ising Model

For non-Ising'ers: Given a graph, we study the probability-distribution on the set of colorings ("Spin-up" and "-down") generated by a given correlation ("force to equality") between adjacient nodes ...

**3**

votes

**2**answers

244 views

### Question on Sparse Random Graphs

I saw stated in a paper the following result but without a reference or a proof.
Let $G$ be an Erdos-Renyi random graph with $n$ nodes and probability of connection $c/n$ with $c>1$. Let $H$ be ...

**1**

vote

**1**answer

245 views

### Connectivity of a graph with fixed number of vertices and edges

Hi,
first of all I want to mention, that I'm pretty new to graph-theory. Currently I'm about to write a path search algorithm and I want to take advantage of previous knowledge.
So this is the ...

**5**

votes

**1**answer

284 views

### A more efficient way to generate random graphs with a given degree sequence?

In graph data mining it is often useful to generate random (simple) graphs with a given degree sequence (e.g. in searching for network motifs), and ideally these would be uniformly at random.
[For ...

**3**

votes

**2**answers

295 views

### Can you randomly sample graphs with quadratic growth?

Let $\mathcal{G}$ be the set of all infinite connected graphs with the following properties:
Every vertex has $4$ neighbors
For every vertex, there are $8$ vertices that have distance exactly $2$ ...

**6**

votes

**3**answers

1k views

### Random bipartite graphs

Consider the following situation: I have a set $A$ of $n$ vertices and a set $B$ of $N = n^2$vertices. I consider the bipartite graph $(A, B)$ and put at random $M = n^{1 + \varepsilon}$ edges (or I ...

**7**

votes

**2**answers

900 views

### Probability of Generating a Connected Graph

$N$ points are generated randomly within a unit square, with a uniform distribution.
What is the probability that the points form a connected graph, given that two points are connected if the distance ...

**0**

votes

**1**answer

263 views

### How many graphs with given average degree and average number of outgoing nodes?

Hi,
does anyone know if it is known what is the number of undirected graphs with the following properties:
Number of nodes: $N$, a big number,
Average degree: $z_1$,
Average number of outgoing ...

**8**

votes

**1**answer

447 views

### Change in the average geodesic distance of a graph when flipping a single edge

Is there a way to determine how the average geodesic distance between nodes of a graph will change just by flipping (1) a single edge without having to traverse the whole graph like in the Djikstra ...

**6**

votes

**4**answers

240 views

### Finite graphs that realize all types over $n$-element sets

Call a graph $G$ $n$-saturated if for every set $A$ of size $n$ of vertices and all $B\subseteq A$ there is a vertex $v\not\in A$ that forms an edge with all $w\in B$ and
does not form an edge with ...

**1**

vote

**1**answer

184 views

### PR[$\lambda_2 > x$] in $G_{np}$ model

Hi!
Does anyone know of any statement relating the probability that the second largest eigenvalue of a random graph is bigger than $x$ to the parameter $p$ in the $G_{np}$ model?

**9**

votes

**2**answers

719 views

### Properties of Some Random Graphs

Working in a problem the following family of graphs appears naturally. Consider the set $A_{n}=\{1,2,3,\ldots,n\}$ and let $\mathcal{C_{n}}$ be the set of all permutations of $A_{n}$ of order $n$ ...

**4**

votes

**2**answers

931 views

### Probability, that a graph G does not contain a cycle

Hello, given graph $G=(V,E)$ with $n=|V|$ and $k=|E|$, what is the probability that it does not contain any cycle $C_l$ for $l\geq3?$
The requested clarification:
My intention was to form the ...

**7**

votes

**4**answers

1k views

### “sparse graphs are locally tree-like”

I would like to be able to state with confidence that sparse graphs (graphs with small numbers of edges) are locally tree-like (they have few short cycles). Apparently "Sparse graphs are locally tree ...