# Tagged Questions

**3**

votes

**0**answers

147 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

396 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

201 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

337 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

159 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

55 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 ...

**0**

votes

**0**answers

93 views

### Adjacent matrix of undirected graph with a giant component

Assuming there is a undirected random graph $G=(V,E)$, $|V|=N$ and its adjacent matirx is $A$.
What is the sufficient and necessary conditions of A for that there is a giant component of graph $G$?
...

**3**

votes

**0**answers

108 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

342 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

159 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 ...

**0**

votes

**0**answers

57 views

### problem about Matrix multiple

Assuming a series of random geometric graph G1,G2,...,Gn. For each Gi, |V|=N, and nodes are distributed as a possion point process. If the distance between a pair of nodes is less than or equal to ...

**2**

votes

**0**answers

81 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

212 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

55 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

223 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

459 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

158 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

63 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

92 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

225 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

221 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

162 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

283 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

119 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 ...

**3**

votes

**1**answer

223 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

192 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

152 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 ...

**5**

votes

**0**answers

225 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

571 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

238 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

237 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

269 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

292 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

881 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

260 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

423 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

236 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

181 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

693 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

891 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 ...

**1**

vote

**2**answers

823 views

### Max cut value in a random graph

Let $G = G(n, 1/2)$ be an Erdos-Renyi graph in which each edge $e = (u,v)$ is present in the graph independently with probability $1/2$. For a subset of the vertices $S$, the cut value $c(S)$ is equal ...

**11**

votes

**3**answers

586 views

### Rainbow matchings (in random graphs)

Suppose we have an $(n,n)$-bipartite graph with edges colored with $k$ colors. Is anything known about the existence of rainbow matchings (i.e. a matching that uses each color exactly once, for $k=n$) ...