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Graph Theory - Laplacian/Combinatorial Laplacian/Normalized Laplacian
Formal definition of Laplacian is as follows. In a Graph G, Laplacian L is defined as
Let's assume that we have a Graph as shown below.
The adjacency matrix of this graph is as follows (try to build this matrix on your own as a practice).
The degree matrix of this graph is as follows (try to build this matrix on your own as a practice).
The Laplacian of this graph calculated from 'D-A' become as follows.
In many case, 'Normalized Laplacian' are used. The Normalized Laplacian is defined as follows. (As you see, we divide all the elements of Laplacian in such a way that the diagonal values become '1'. (di is the diagonal values on Laplacian matrix).
Normalized Laplacian given in this example become as follows :