  In this article we consider the spectrum of a Laplacian matrix, also known as
the Markov matrix, under the independence assumption. We assume that the
entries have a variance profile. Motivated by recent works on generalized
Wigner matrices we assume that variance profile gives rise to a sequence of
graphons. Under the assumption that these graphons converge, we show that the
limiting spectral distribution of the scaled Laplacian converge. We give an
expression for the moments of the limiting measure in terms of graph
homomorphisms. In some special cases we identify the limit explicitly. We also
study the spectral norm and derive the order of the maximum eigenvalue. We show
that our results cover Laplacian of various random graphs include inhomogeneous
Erd\H{o}s- R\' enyi random graph, sparse W-random graphs, stochastic block
matrices and constrained random graphs.
