  Symmetric matrices with zero row sums occur in many theoretical settings and
in real-life applications. When the offdiagonal elements of such matrices are
i.i.d. random variables and the matrices are large, the eigenvalue
distributions converge to a peculiar universal curve
$p_{\mathrm{zrs}}(\lambda)$ that looks like a cross between the Wigner
semicircle and a Gaussian distribution. An analytic theory for this curve,
originally due to Fyodorov, can be developed using supersymmetry-based
techniques.
  We extend these derivations to the case of sparse matrices, including the
important case of graph Laplacians for large random graphs with $N$ vertices of
mean degree $c$. In the regime $1\ll c\ll N$, the eigenvalue distribution of
the ordinary graph Laplacian (diffusion with a fixed transition rate per edge)
tends to a shifted and scaled version of $p_{\mathrm{zrs}}(\lambda)$, centered
at $c$ with width $\sim\sqrt{c}$. At smaller $c$, this curve receives
corrections in powers of $1/\sqrt{c}$ accurately captured by our theory. For
the normalized graph Laplacian (diffusion with a fixed transition rate per
vertex), the large $c$ limit is a shifted and scaled Wigner semicircle, again
with corrections captured by our analysis.
