All Questions
75 questions
0
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56
views
Does Forcing conjecture equals to assume the host graph is regular?
Given two graphs $H$ and $G$, the homomorphism density $t(H, G)$ is defined as the proportion of mappings from the vertices of $H$ to the vertices of $G$ that preserve adjacency. Formally,
$$
t(H, ...
- 349
0
votes
0
answers
45
views
Another version of Sidorenko's conjecture(?)
I would like to ask a question about Sidorenko's conjecture. Here is the background of my question:
Quasi-random graphs
A sequence of graphs $(G_n)$ is called quasi-random if it satisfies certain ...
- 349
3
votes
0
answers
81
views
Can we remove the restriction on a parameter in Talagrand concentration inequality?
Recently I am trying to use Talagrand concentration inequality to do something on graphs. I find a version from the book of Molloy and Reed ''Graph Colouring and Probabilistics Method''. I attached a ...
- 1,190
2
votes
1
answer
80
views
Positive-semidefiniteness of Laplacian of signed graph
Consider a signed complete graph $G(E,V)$ with adjacency $A_{ij}\in\{-1,+1\}$. Define the Laplacian matrix as $L:=D-A$ where $D$ is the degree matrix, $D_{ii}=\sum_{j\neq 1}A_{ij}$.
my question.
If $\...
- 405
0
votes
0
answers
52
views
Does "epsilon-regular" equal to "cut distance less than epsilon"?
Let $G$ be a bipartite graph (vertex number sufficient large) with bipartition $(U,W)$ and edge density $d$. Does these two statement equal?
$G$ is $\varepsilon$-regular, i.e. $\big|e_G(X,Y)-d|X||Y|\...
- 349
1
vote
0
answers
99
views
Szemeredi Regularity Lemma - Reasonable Bounds
Recently, I came across the wonderful Szemeredi regularity lemma and I was wondering. Are these "natural/reasonable/standard" examples of random graphs families, for which the number of $\...
- 5,405
1
vote
0
answers
164
views
Locally "unshortable" paths in graphs
Setup: Consider a connected graph G, with diameter "d".
Informally: Trivially (by definition of diameter), taking any path $P$ any nodes $P(i) , P(i+k)$ for $k>d$ can be connected by a ...
- 24.9k
3
votes
0
answers
151
views
Smallest dominating set
Given a graph $G$, we say $S$ is a dominating set if $S\cup \{N(x):x\in S\}=V(G)$. Let $d(n,k)$ be the smallest integer $s$ so that every $n$-vertex graph $G$ with minimum degree $k$ has some ...
- 3,499
2
votes
2
answers
286
views
Finding an easy example applying the general Lovász local lemma
Is there any easy application for the general local lemma as follows? If someone knows, please tell me the references or just post an example here. Thanks.
General Lovász local lemma: Consider a set $...
- 1,190
2
votes
0
answers
91
views
Graphon convergence of uniform weighted graphs
I have a question that I need at some point my research. Suppose that the upper-triangular entries of an $n\times n$ symmetric matrix $A$ are i.i.d. Uniform$(0,1)$. Does the weighted graph with ...
- 241
5
votes
1
answer
319
views
Probability of the random graph on $2n$ vertices having exactly $n$ vertices with degree $\ge n$
Let $G = (V, E)$ be a uniform random graph on $2n$ labeled vertices and let $S \subseteq {V}$ be the set of vertices with degree $\ge n$. Then what happens to $\mathbf{P}(|S|=n)$ as $n \to \infty$?
...
8
votes
1
answer
392
views
What is this Ramsey problem?
Given positive integers, $n,m,r$, define $R((n,m);r)$ to be the least $N$ such that for any $r$-coloring $C:E(K_N)\to \{1,\dots,r\}$, there is some monochromatic subgraph with $n$ vertices and $m$ ...
- 3,499
1
vote
0
answers
115
views
Probability of (single) connecting paths in Erdos-Renyi graphs
In an Erdos-Renyi graph with labeled vertices in $(1, ..., N)$, and for any pair of vertices $(r, s)$ with $r < s$ and a length $l$ in $(1, ..., s-r)$, I am looking for the probability of
there ...
- 129
2
votes
1
answer
96
views
Lower bound on the number of balanced graphs
Let $\alpha>1$ be a constant and define $B_n$ as the number of (labeled) balanced graphs with $n$ vertices and $\left\lceil \alpha n\right\rceil $ edges. The paper Strongly Balanced Graphs
and ...
- 143
6
votes
0
answers
164
views
Hamilton cycles in random graphs with just enough connectivity
What is the asymptotic probability that $G$ has a Hamilton cycle if $G$ is a random $n$ vertex $\frac{4}{3}n$ edge graph, with minimum degree 2 and without degree 2 vertices at distance 1 or 2 to each ...
- 7,545
6
votes
2
answers
723
views
Threshold function for a graph not being planar
A graph property $\mathcal{P}$ is monotone increasing if $G\in \mathcal{P}$ implies $G+e \in \mathcal{P}$, i.e., adding an edge to a graph does not destroy the property.
It is well-known that every ...
- 557
3
votes
1
answer
163
views
The most pseudorandom subgraph of a dense graph
A bipartite graph $(A,B)$ is $(p, \beta)$-jumbled if for all subsets $A'\subseteq A$ and $B'\subseteq B$ we have that $\left|\mathrm{E}(A',B')-p|A'||B'|\right|\leq \beta \sqrt{|A'||B'|}$. A easy ...
- 805
2
votes
1
answer
165
views
Is a random $(r+1,r)$-biregular bipartite graph $r$-edge connected w.h.p?
A uniformly random $r$-regular bipartite graph on $n$ vertices is known to be $r$-edge connected. That is, with high probability as $n$ grows large, the minimum size of a cut in a random $r$-regular ...
- 153
3
votes
0
answers
148
views
Random graph - probability threshold for any linear size set to contain a fixed clique
Let $t\geq 3$ and $0<\varepsilon<1$ be fixed. Denote by $K_t$ the clique on $t$ vertices, and by $G_{n,p}$ the binomial random graph.
Question:
Is the threshold for the probability that "...
1
vote
1
answer
119
views
Does exponential degree distribution entail Log-normal distance distribution in large complex graphs?
We've been exploring the graph structure of a large genealogical data base (WikiTree) of which main connected component contains about 23 million nodes. The graph edges are defined by any direct ...
2
votes
3
answers
230
views
Random graphs defined by a set of tiles
Related to this question, which I asked at MSE, I'd like to ask this one here:
Consider a (large) graph $G$ and its multi-set of tiles $T$, i.e. the multi-set of its vertex-induced subgraphs, i.e. the ...
- 9,720
1
vote
0
answers
40
views
Eigenvalue bounds of a random graph with a clique
I'm looking into this paper and having some problems proving (ii) of proposition 2.1. I don't quite understand how the lemma is proved. I also read the original paper where the lemma comes from but ...
- 11
2
votes
0
answers
282
views
Generating a random graph with bounds on degree and diameter
What would be a way to generate a random simple graph with diameter lesser than a given number, and in which there are given lower and upper bounds (bounds being uniform across vertices) on the degree ...
- 1,216
23
votes
4
answers
978
views
What nodes of a graph should be vaccinated first?
Consider a graph, choose some "p: 0<p<1" (probability to infect the neighbor node).
Choose some random number "K" of nodes which are "infected" initially.
So we ...
- 24.9k
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
5
votes
0
answers
191
views
Additional examples of classes of networks whose Hasse diagram of the poset is a perfect graph
This question is very important for my research, which is why I ask it here.
I do not have a formal background in graph theory so please excuse me if I state a term incorrectly (and feel free to ...
- 51
3
votes
1
answer
164
views
making a random uniform hypergraph linear
Let $\mathcal{H}_{n,p,h}=(V,E)$ be a random $h$-uniform hypergraph on $[n]$, sampled according to the usual binomial distribution. We known that with high probability, the number of edges in $\mathcal{...
3
votes
1
answer
159
views
Hyper-degree sequences: How to count them and how to construct hyper-graphs from them?
From an answer to this question I have learned how to ask this question properly.
Consider a $k$-uniform hypergraph on $n$ nodes, i.e. a family of $k$-subsets of $[n]= \{1,2,\dots,n\}$ (the hyperedges)...
- 9,720
3
votes
0
answers
209
views
Two kinds of generating functions
Sorry for a possibly off-the-topic question, but I am afraid to gain the necessary overview to give an answer (supposed the question is not ill-posed) is beyond my capabilities.
In the course of ...
- 9,720
3
votes
1
answer
138
views
Degree sequences after vertex removals
Consider a graph $G=(V, E)$ with $|V|=n$ vertices. Let $(v_1, \dots, v_n)$ be an ordered list of its vertices. Let $G_i=G[\{v_{i+1}, \dots, v_n\}]$ be the induced subgraph on the last $n-i$ vertices. ...
- 565
186
votes
3
answers
96k
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Issue UPDATE: in graph theory, different definitions of edge crossing numbers - impact on applications?
QUICK FINAL UPDATE: Just wanted to thank you MO users for all your support. Special thanks for the fast answers, I've accepted first one, appreciated the clarity it gave me. I've updated my torus ...
- 343
11
votes
1
answer
370
views
Graph with path of length $\geq n$ along grid diagonals - a known result in graph theory?
Is the following lemma a well known result in graph theory?
I am studying a basic existence result that appears to be simple yet powerful. I have not seen it stated as an important result in graph ...
- 6,917
1
vote
0
answers
140
views
Count shortest path with different lengths in random graph
Let $G(n,p)$ be an Erdos-Renyi random graph on $n$ vertices with probability $p$, i.e. for each pair of vertices, they are connected directly by an undirected edge with probability $p$. Suppose we are ...
- 1,495
1
vote
1
answer
436
views
Size of minimum cut in random graph
Consider a uniform random tournament with $n$ vertices. (Between any two vertices $x,y$, with probability $0.5$ draw an edge from $x$ to $y$; otherwise draw an edge from $y$ to $x$.) The score of each ...
- 1,209
2
votes
0
answers
83
views
Zero-One law for Hamiltonian path subgraphs of Hamming Distance Graphs?
$(\alpha,\beta,d)$-Hamming Distance Graph $G_d(\alpha,\beta)$ for $\alpha,\beta\in(0,1]$ is a graph on $2^d$ vertices $v_0,\dots,v_{2^d-1}$ with edges $(v_i,v_j)\in\mathcal E(G_d)$ iff $0<\sum_{t=1}...
- 13.9k
3
votes
1
answer
598
views
Asymptotic formula for the number of connected graphs
It can be shown that the set of graphs with $N$ vertices $G_N$ has cardinality:
\begin{equation}
\lvert G_N \rvert = 2^{N \choose 2} \tag{1}
\end{equation}
Recently, I wondered how much bigger $\...
- 3,871
1
vote
2
answers
116
views
How to use probability to find a matching in a family of graphs?
In a conference, I heard that we can use some probabilistic methods to find a matching in some kind of graphs. I would like to see some examples of such technics. Can someone provide some references ...
- 57
2
votes
2
answers
536
views
Modularity in a graph -- derivation of modularity score
Background
I am currently reading "Modularity and community structure in networks" (2006) by Newman [1].
In it, he derives a score for the modularity of a graph ...
- 121
3
votes
1
answer
108
views
Expected size of matchings in a cubic graph
Let $G$ be a random cubic graph on $n$ vertices. Let $M$ be the set of (not necessarily maximum) matchings of $G$. What is the expected size (i.e. number of edges) of an element of $M$?
In other ...
- 9,501
3
votes
1
answer
822
views
Open Problems in Random Graphs [closed]
I am a PhD student in mathematics. I'm interested in probabilistic methods in combinatorics and especially random graphs. I am looking for an open problem in this area for my PhD proposal. I know that ...
- 57
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
12
votes
3
answers
1k
views
A Modern Proof of Erdos and Renyi's 1959 Random Graph Paper?
In their paper, Erdos and Renyi consider a random graph with a fixed number of edges, as opposed to the more modern approach of adding each edge independently with probability $p$. From what I ...
- 470
2
votes
1
answer
607
views
Component size distribution in small Erdos-Renyi networks
I'm looking at $\mathcal{G}(n,p)$ (I'll call these Erdos-Renyi networks) where $n$ is, say, at most 10.
I would like to know the probability a random node is in a component of size $m$.
It's ...
- 121
2
votes
0
answers
49
views
Size of the last non-empty $k$-core of a random graph
Given $n$ and $p$ for $G(n,p)$, how to find the distribution of the size of the non-empty $k$-core with largest $k$?
In particular, what is the probability (for any $n$ and $p$) that only $c$ ...
- 21
1
vote
1
answer
188
views
KPZ relation $\chi = 2 \xi -1$ in a random geometric graph
If I have $n$ points uniformly distributed on the surface of a torus, and form a graph by adding an edge between pairs whenever they are within a unit distance (induced by the Euclidean metric), I ...
- 640
6
votes
1
answer
341
views
Bounds on degrees of minors obtained by edge contractions of regular graphs
Given a connected $d$-regular graph $G=(V,E)$, generate a sequence of minors by performing only edge contractions and loop deletions (as, e.g., in Karger's algorithm) until the graph collapses to a ...
- 163
1
vote
1
answer
1k
views
An explicit formula for the number of different (non isomorphic) simple graphs with $p$ vertices and $q$ edges
I would like to know if there is an explicit formula for the number of different (non isomorphic) simple graphs with a given number of vertices $p$ and edges $q$, and if yes what is it.
Trying to ...
- 29
13
votes
1
answer
409
views
When is the union of a graph and a random permutation thereof connected?
First things first: in what follows, a "random permutation" of a set $\Omega$ with $n$ elements does not necessarily mean an element chosen uniformly at random from $\textrm{Sym}(\Omega)$. Rather, and ...
- 20.2k
0
votes
1
answer
1k
views
Random graphs- Erdos and Renyi 1959 paper
Please refer to this link. It is Erdos and Renyi's first paper on Random Graphs (1959). I am trying to work through it.
I'm struggling with equations (16), (17) and (21).
(16)
I'm not sure why ...
0
votes
1
answer
3k
views
How to compute the clustering coefficient of a random graph?
How is the clustering coefficient defined for random graphs? For example, a first definition could be calling clustering coefficient of a random graph the expected value of the clustering coefficient ...
- 23