All Questions
Tagged with random-graphs spectral-graph-theory
20 questions
1
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0
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91
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Diameters of random bipartite graphs
Given two partite sets of vertices $U$ and $V$ of size $n$. Each vertex in $U$ uniformly randomly selects $K$ ($K$ is a constant and $K\ll n$) vertices in $V$ without replacement and connects a ...
- 11
1
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54
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Controlling quantity related to Laplacian pseudo-inverse of Erdős–Rényi graph
Consider an $n$-node undirected graph $G = (V, E)$ equipped with weights $W$. Let $L$ be the weighted graph Laplacian matrix, i.e. $L_{ij} = -W_{(i,j)}$ for $(i,j)\in E$ and $L_{ii} = \sum_{j:(i,j)\in ...
- 11
1
vote
1
answer
543
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Vertex degree on random graphs
Let $p = d/n$ with $d$ constant. How do I prove that, with high probability, $G_{n,p}$ contains a vertex of degree at least $(\log n)^{1/2}$,
where $G_{n,p}$ is a graph with $n$ vertices and the ...
- 21
1
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64
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Angles between edges of a geometric graph and graph invariants
Are there any clever ways in which the angles between edges in a geometric graph are encoded in the graph spectrum, or another object associated with the graph?
I'm interested to see what else is ...
- 640
2
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1
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128
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Are the eigenvalues of the 1D lattice with random weights known?
Consider the adjacency matrix $\mathbf{A}$ of a one dimensional lattice of size $N$. That is, $A$ is a $N\times N$ matrix with $A_{ij}=1$ if vertex $i$ adjacent to vertex $j$ (there exists an edge ...
- 274
2
votes
1
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211
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The complexity of expansion ratio (Cheeger constant) of a graph
Let $G=(V(G), E(G))$ be a graph on $n$ vertices and let $S$ be a subset of $V(G)$. The boundary of $S$, denoted by $\partial S$, is the set of edges $(i, j)$ such that $i \in S$ and $j \in V(G) \...
- 640
2
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212
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Expansion of random subgraphs of a bi-regular bipartite graph
Let $G = (L, R, E)$ be a bi-regular bipartite graph, with $|L|=n$ and $|R| = C \cdot n$, where $C$ is a large constant. Let $d$ be its (constant) right-degree.
We know $G$ is a good spectral expander. ...
- 225
0
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0
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101
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4-cycles vs eigenvalue information on quasi-random graphs
My (philosophical) question arises from reading the wonderful paper of Chung-Graham-Wilson where the authors introduces the notion of quasi-random graphs.
The main purpose of the paper is to show ...
- 1,561
1
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0
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61
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$\exp(-Cn^{\epsilon})$ estimate for probability of Brouwer-Haemers condition in Erdos-Renyi-like random graph
For any $n$-vertex graph $G$, we have the inequality $\lambda_i^{L_G}\geq D_i-i+2,$ where $L_G$ denotes the Laplacian of $G$ and $\lambda_i^{L_G}$ denotes the $i^\text{th}$ largest eigenvalue and $D_i$...
- 276
1
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0
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209
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Characterization of k-walk-equivalent graphs
Let $G=(V,E)$ be an undirected graph. A walk of length $k$ in $G$ is a sequence of vertices $v_1,v_2,\ldots,v_{k+1}$ in $V$ such that $(v_i,v_{i+1})\in E$ for each $i=1,2,\ldots,k$.
Call two graphs $...
- 493
4
votes
1
answer
412
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Distribution of eigenvectors and eigenvalues for random, symmetric matrix
Consider two simple, undirected graphs with adjacency matrices ${\bf A}$ and ${\bf A'}$. Let ${\bf P} = {\bf A'} - {\bf A}$. Thus, ${\bf P}$ is symmetric and always 0 along the diagonal.
Let $f({\bf ...
- 43
2
votes
2
answers
110
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Volume doubling implies that the degree is uniformly bounded above?
Let $G=(V,E)$ be a connected graph. Here $V$ is the set of all vertices of $G$, and $E$ is the set of all edges of $G$. Suppose that $G$ is locally finite, i.e., $\sharp\{y\sim x:y \in V \}$ is finite ...
- 169
12
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1
answer
526
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Spectrum of induced subgraphs of Paley graph
Let $G_q$ be a Paley graph on $q$ vertices, where $q=1 \text{ (mod 4)}$, i.e., the vertices of $G_q$ are the elements of the finite field $\mathbb{F}_q$, and there is an edge between vertices $a,b \in ...
- 171
5
votes
2
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845
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Database of adjacency matrices on cospectral non-isomorphic graph pairs
Is there a repository of cospectral non-isomorphic graphs available somewhere?
I am looking for list of $0/1$ adjacency matrix pairs that can be input data in tools such as MATLAB.
- 13.9k
8
votes
2
answers
702
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limiting empirical spectral distribution of the Laplacian matrix on an Erdos-Renyi graph?
Let $G$ be an Erdos-Renyi random graph (i.e. an edge ($ij$) exists with probability $0 < p < 1$ and all edges are independent). Let $L$ be the Laplacian matrix of this graph (i.e $L=D-A$, where $...
5
votes
1
answer
1k
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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 ...
- 6,962
10
votes
3
answers
4k
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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 ...
- 2,449
10
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3
answers
1k
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Eigenfunctions of random graphs
Consider a random $d$-regular graph on $n$ vertices. What can be said about its nontrivial (i.e. orthogonal to the constant) eigenfunctions? For example, I'm interested whether there are "nodal zones",...
- 2,449
1
vote
1
answer
223
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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?
- 239
8
votes
2
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986
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Spectrum of the Laplacian on G(n, p) and G(n, M)
A random graph in $G(n, p)$ model is a graph on $n$ vertices in which for each of the $n\choose{2}$ edges we independently flip a coin, then take the edge with probability $p$ or remove it with $1 - p$...
- 2,449