How networks with high largest eigenvalues are more robust? In the literature, it is sometimes indicated that network with high value of largest eigenvalue (either adjacency matrix or its Laplacian counterpart) are more robust against link/node removals. However, these statements are usually NOT accompanied with references. 
I am looking for some explanation on why is this so? Or pointers to some work that investigate/explain this.
I wonder if this statement is simply the result of difference between link density? I doubt its so simplistic. Is there any study on network robustness comparing networks with same link density but different largest eigenvalues (or spectral radius)?
 A: I am not sure, because no specific reference was given, but I suspect this is referring to the well known fact that the isoperimetric constant can be bounded below by eigenvalues, as in Proposition 4.2.5 of Lubotzky's book Discrete Groups, Expanding Graphs and Invariant Measures.  If the isoperimetric constant is $c$, then a set of $m$ nodes cannot be isolated by removing less than $cm$ links (unless the set has more than half of the vertices in the entire graph).
A: As @Morris answered the reason is behind in connectivity and rapid connection which is compacted in isoperimetric parameter of graphs. The isoperimetric parameter has bounded by eigenvalues in some nice inequality.  Also, you can see that Cayley graphs are good objects for designing network, which one of main reason is its connectivity, since an $r$-regular connected cayley graph has connectivity $r$. But for study much more I suggest these two books which are very good in related to your question:
"Graph Spectra for Complex Networks" by Piet van Mieghem
"Expander Families and Cayley Graphs: A Beginner's Guide" by Mike Krebs and Anthony Shaheen.
A: The following papers might contain relevant information to your question:
Robustness of networks against viruses: the role of the spectral radius, by
A. Jamakovic, R.E. Kooij, P. Van Mieghem, E.R. van Dam
https://www.nas.ewi.tudelft.nl/people/Rob/telecom/jamakovic.pdf
Epidemic Spreading in Real Networks: An Eigenvalue Viewpoint, by 
Yang Wang, Deepayan Chakrabarti, Chenxi Wang and Christos Faloutsos 
http://www-2.cs.cmu.edu/afs/cs.cmu.edu/user/christos/www/PUBLICATIONS/srds03-virus.pdf
