We study a class of Hermitian random matrices which includes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as adjacency matrices of Erdos-Renyi random graphs with p(n) similar to 1/n. Our n x n random matrices have real entries which are i.i.d. up to symmetry. The distribution of entries depends on n, and we require row sums to converge in distribution. It is then well- known that the limit distribution must be infinitely divisible. We show that a limiting empirical spectral distribution (LSD) exists and, via local weak convergence of associated graphs, that the LSD corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many Poisson weighted infinite trees using a backbone structure of special edges called "cords to infinity". One example covered by the results are matrices with i.i.d. entries having infinite second moments but normalized to be in the Gaussian domain of attraction. In this case, the limiting graph is N rooted at 1, so the LSD is the semicircle law.