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.