3
$\begingroup$

Let $A$ be an $n \times n$ sparse matrix generated via the Erdős-Rényi method. Here, "sparse" means that $\|A\|_F = O(n)$. I am interested in the relationship between the expectation $\mathbb E(\rho(A))$ and the Erdős-Rényi density $p$ when $n$ is large.

Using MATLAB, I find that when $A$ is sparse (i.e., the density is small), the expectation of the spectral radius is almost linear in the matrix density. Can this result somehow be proved?

Any reference or suggestion will be helpful.

Improve this question
$\endgroup$
1
  • $\begingroup$ Do you mean the adjacency matrix of a random E-R graph? $\endgroup$
    – Igor Rivin
    Commented Nov 14, 2017 at 5:52

1 Answer 1

Reset to default
3
$\begingroup$

If you mean that $A$ is the adjacency matrix of an Erdos-Renyi random graph, then the question has been studied, and your conjecture is false (but just barely). See

Krivelevich, Michael; Sudakov, Benny, The largest eigenvalue of sparse random graphs, Comb. Probab. Comput. 12, No.1, 61-72 (2003). ZBL1012.05109.

(available on arxiv.org).

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .