I have large $(0-1)$-matrices with the additional property that the first row and the first column are almost all $1$'s. My question is

Is there a (good) algorithm to obtain an upper bound for the largest eigenvalue of such a matrix ?

This question actually didn't come up in my own research but some collaborator of mine asked this question at dinner these days. I thus don't know more about the background of the question, but my impression would be that if anything is known, this is the right place to ask.

Thanks, Christian