Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have a matrix $A \in \{0,1\}^{n \times n}$ -- i.e. a matrix with 1s and 0s only.

Is there a way to find or upper bound its largest eigenvalue?

I have a feeling it is related to connectivity of directed graphs, if $A$ is thought of as adjacency matrix.

Also, when trying

      A = rand(10,10); A = A <= p; [x,S] = eig(A+0.00); S(1,1)

in Matlab for various values of p (0.1,0.2,...) (many times) it seems like S(1,1), the largest eigenvalue, is around 1/p.

Any help is appreciated.

share|improve this question
1  
You may find this blog post relevant: qchu.wordpress.com/2009/04/30/… –  Brad Sep 29 '12 at 16:06
    
Also try link –  Patrick Li Sep 29 '12 at 16:07
    
@PatrickLi thanks, that's useful. Brad: couldn't find exactly an answer there. –  kloop Sep 29 '12 at 16:39
    
You should symmetrise $A \mapsto (A+A')/2$ to get adjacency matrices... –  draks ... Nov 7 '12 at 21:14
add comment

1 Answer

The largest eigenvalue of the adjacency matrix of a graph is bound by $$ \delta(A)\le\mu_{\text{max}}(A)\le\Delta(A), $$ with $\delta$($\Delta$) being the smallest(largest) vertex degree in your graph. A proof can be found here(Lemma 3). In your case $\delta$($\Delta$) lie around $\frac1p$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.