TITLE: What is the largest possible operator norm of a sparse (0,1)-matrix?
QUESTION [5 upvotes]: Inspired by this question, I was wondering about the following problem:
Consider all $n\times n$ $(0,1)$-matrices with $k$ ones.  Which of these matrices has the largest operator norm?  And how does this largest possible operator norm behave with $n$ and $k$?
If any progress has already been made on this problem, I assume it's in the guise of directed graphs.  For the case of symmetric matrices with zero diagonal, which graphs give the largest possible largest eigenvalue?  $(k/n)$-regular graphs?

REPLY [4 votes]: If $k=ab$ with $a\leq n$ and $b\leq n$, then an $a \times b$ rectangle of $1$s surrounded by $0$s hits the optimal matrix norm of $\sqrt{k}$.  Otherwise, $\sqrt{k}$ is not achievable, but it's clear that one can get very close to $\sqrt{k}$ with very close to a rectangle.
A $k/n$-regular graph has a largest eigenvalue of $k/n$ which is not that great. A better thing to do is a complete graph on about $\sqrt{k}$ vertices plus $n-\sqrt{k}$ isolated vertices, which gives a largest eigenvalue of about $\sqrt{k}$.