  Let $M_n$ be a class of symmetric sparse random matrices, with independent
entries $M_{ij} = \delta_{ij} \xi_{ij}$ for $i \leq j$. $\delta_{ij}$ are
i.i.d. Bernoulli random variables taking the value $1$ with probability $p \geq
n^{-1+\delta}$ for any constant $\delta > 0$ and $\xi_{ij}$ are i.i.d.
centered, subgaussian random variables. We show that with high probability this
class of random matrices has simple spectrum (i.e. the eigenvalues appear with
multiplicity one). We can slightly modify our proof to show that the adjacency
matrix of a sparse Erd\H{o}s-R\'enyi graph has simple spectrum for
$n^{-1+\delta } \leq p \leq 1- n^{-1+\delta}$. These results are optimal in the
exponent. The result for graphs has connections to the notorious graph
isomorphism problem.
