The Laplacian Matrix of a graph is a symmetric matrix having the same number
of rows and columns as the number of vertices in the graph and element (i,j)
is d[i], the degree of vertex i if if i==j, -1 if i!=j and there is an edge
between vertices i and j and 0 otherwise.
A normalized version of the Laplacian Matrix is similar: element (i,j) is 1
if i==j, -1/sqrt(d[i] d[j]) if i!=j and there is an edge between vertices i
and j and 0 otherwise.
The weighted version of the Laplacian simply works with the weighted degree
instead of the plain degree. I.e. (i,j) is d[i], the weighted degree of
vertex i if if i==j, -w if i!=j and there is an edge between vertices i and
j with weight w, and 0 otherwise. The weighted degree of a vertex is the sum
of the weights of its adjacent edges.