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3 votes
1 answer
336 views

Eigenvalues of random graphs

At time $t=0$, let $G_n(V,E)$ be a graph with $n$ vertices and $m < n$ edges. Then there exists a unique symmetric adjacency matrix $A_n$ associated with $G_n(V,E)$, defined as follows: $a_{ij} = 1$...
Piero Giacomelli's user avatar
3 votes
1 answer
454 views

Difference of pseudoinverse bound under assumptions

This seems like a standard problem, but unable to find a solution online. Suppose we have two singular PSD matrices A and B with the following assumptions: $ 0 < x \leq ||A|| \leq y$ $ 0 < ||...
MJagota's user avatar
  • 33
2 votes
0 answers
64 views

Largest eigenvalue of two types of slightly different random matrices

Consider two types of slightly different $n \times n$ symmetric random matrices $X$. The diagonal elements of $X$ are fixed as $1$. Suppose $\frac{k}{n} \to \alpha$ for some constant $\alpha\in(0,1)$. ...
Tony's user avatar
  • 272
3 votes
2 answers
579 views

Largest eigenvalue of the adjacency matrix of weighted random graph

I find the theorem for largest eigenvalue of the adjacency matrix of ER random graph in here https://arxiv.org/pdf/math/0106066.pdf. The adjacency matrix is a symmetric random matrix s.t. diagonal ...
Tony's user avatar
  • 272
3 votes
0 answers
151 views

Largest eigenvalue divided by $n$

Let $X$ be an $n\times n$ symmetric random matrix whose diagonal is fixed as $1$, and every element in the upper triangle (excluding the diagonal) is drawn from Bernoulli($p$). The elements in the ...
Tony's user avatar
  • 272