  We consider the spectrum of random Laplacian matrices of the form
$L_n=A_n-D_n$ where $A_n$ is a real symmetric random matrix and $D_n$ is a
diagonal matrix whose entries are equal to the corresponding row sums of $A_n$.
If $A_n$ is a Wigner matrix with entries in the domain of attraction of a
Gaussian distribution the empirical spectral measure of $L_n$ is known to
converge to the free convolution of a semicircle distribution and a standard
real Gaussian distribution.
  We consider real symmetric random matrices $A_n$ with independent entries (up
to symmetry) whose row sums converge to a purely non-Gaussian infinitely
divisible distribution, which fall into the class of L\'evy-Khintchine random
matrices first introduced by Jung [Trans Am Math Soc, \textbf{370}, (2018)].
Our main result shows that the empirical spectral measure of $L_n$ converges
almost surely to a deterministic limit. A key step in the proof is to use the
purely non-Gaussian nature of the row sums to build a random operator to which
$L_n$ converges in an appropriate sense. This operator leads to a recursive
distributional equation uniquely describing the Stieltjes transform of the
limiting empirical spectral measure.
