recently, wang, chakrabarti, wang and faloutsos have shown that the spectral radius of a graph plays an important role in modeling virus propagation in networks. this led van dam and kooij to consider the following problem: which connected graph on n nodes and diameter d has minimal spectral radius ? van dam and kooij answered this question for $d=n-1,n-2,n-3,n/2,2,1$ and provided a conjecture for the case $d=n-e$, when e is fixed. in this talk, i give an overview of their work and i will outline a proof of their conjecture for $e=4$ and possible extensions for $e>4$. this is joint work in progress with edwin van dam (university of tilburg, the netherlands).