Laplacian Matrix

up:: Adjacency Matrix

Given a Simple Graph with adjacency matrix $A$ and a diagonal matrix of Node Degrees $D$, the Laplacian matrix is defined as

$$L=D-A$$

Properties

Note that, per definition,

$$\begin{array}{rl}\sum _i{L}_{ij}& =\sum _i{D}_{ij}-\sum _i{A}_{ij}\\ & =deg(i)-deg(i)\\ & =0\end{array}$$

Thus, it is equivalent to the adjacency matrix description of the graph $G$, since one can infer its adjacency matrix from the off-diagonal elements, and since the diagonal elements (i.e. node degrees) are equal to the row-wise sum of $L_{i\ne j}$. In other words, it has no more information than the adjacency matrix $A$. Nevertheless, it is useful in some applications.

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References

• Laplacian matrix - Wikipedia