Geometric intuition of graph Laplacian matrices I am reading about Laplacian matrices for the first time and struggling to gain intuition as to why they are so useful. Could anyone provide insight as to the geometric significance of the Laplacian of a graph? For example, why are the eigenvectors of a Laplacian matrix helpful in interpreting the corresponding graph?

Perhaps the most natural definition is the (oriented) incidence transformation.
The incidence transformation of an edge $(u,v)$ is defined by $E(u,v) = v - u$.
Its adjoint $E^*$ has a dual definition, taking vertices to the sum of incoming edges minus the sum of outgoing ones.
The Laplacian is just $L=EE^*$. Thus, for instance, its eigenvalues and eigenvectors are related to the singular values and vectors of the incidence transformation.
