  A dissociation set of a graph is a set of vertices which induces a subgraph
with maximum degree less than or equal to one. The dissociation number of a
graph is the maximum cardinality of its dissociation sets. In this paper, we
study the connected graphs of order $n$ with a given dissociation number that
attains the minimum spectral radius. We characterize these graphs when the
dissociation number is in
$\{n-1,~n-2,~\lceil2n/3\rceil,~\lfloor2n/3\rfloor,~2\}$. We also prove that
these graphs are trees when the dissociation number is larger than $\lceil
{2n}/{3}\rceil$.
