Interpretation of the Laplacian operator in graphs I am studying graphs right now, and I have just discovered the definition of Laplacian operator:
$\Delta:M(G)\longrightarrow Div(G)$ That is given by the formula:
$$\Delta(f)=\sum_{v\in V(G)}\Delta_{v}(f)(v)$$
Where $$\Delta_{v}(f):=deg(v)f(v)-\sum_{e=wv\in E_v}f(w)$$
$\textbf{Remarks}:$
-$G$ is a finite, unweighted multigraph having no loop edges.
-$V(G)$ the set of vertices, $E_v$ set of edges incident to a give vertex $v$
-$M(G)=Hom(V(G),\mathbb{Z})$ and $Div(G)$ the free abelian group on the set $V(G)$
My question is if there is any interpretation of this operator in terms of the “calculus Laplacian” or any interpretation of this operator that gives me some motivation of its uses and relevance on graph theory.
I will appreciate any help, thanks in advance!
 A: First off, this Laplacian operator is connected to the Laplacian operator on functions whose domains are compact sets or manifolds. Indeed, if you take a smooth manifold $M\subset\mathbb R^n$, then one can define the Laplace-Beltrami operator on $M$ as an operator on smooth functions whose domain is $M$. You can show  that if  you  approximate your manifold with a mesh or graph, the  graph Laplacian, properly renormalized, converges to the Laplace-Beltrami operator. That would explain the "Laplacian" name for your operator.
Another way to look at it, is  that your  graph Laplacian applied  to a function $f$ defined on the graph computes the sum  of the differences between the value of $f$ at  each  vertex and the average of  $f$ around that vertex (that  is, on the nearest neighbors). This is very similar to the regular Laplacian, for which the  Taylor formula gives that a smooth function $f$ can be approximated by its average on a sphere of radius $h$ around each point and the error term is proportional to the Laplacian:
$$f(x) = \text{avg}_x(f,h)$ + Ch^2 \Delta  f + o(h^2)$$
When it comes to motivation, a big reason this operator is interesting is that we can learn things about the graph from the properties of the Laplacian (just like we can learn about a manifold by studying the Laplace-Beltrami operator defined on it). More specifically, people have been interested in the eigenvalues and eigenvectors of the operator (it's self-adjoint). For instance, almost 60 years ago, people wondered if the set of eigenvalues of the Laplacian would be a unique signature for a graph or  manifold (see Can you hear the shape of a drum?). While the answer is no, meaning you  could find two distinct  graphs/manifolds with the same Laplacian spectrum, you can still learn a lot about the topology  of a graph from  studying the decay of the eigenvalues.
As importantly, people have looked at the eigenvectors of the Laplacian. It was noticed that eigenvectors and eigenvalues of the  operator are related to diffusion processes on the graph or manifold. And by using those as coordinates, you can embed the graph or manifold into $R^n$ (with a controlled error term if you only use a subset of the eigenvectors). In other words, the eigenvectors of the laplacian can be used as a coordinate system to parametrize points on the graph/manifold. And that parametrization is meaningful as it captures geometric and topological features of the graph/manifold. For instance, it preserves the degree of connectivity of pairs of vertices: Two vertices that are connected by a lot of paths will be projected to close points in the embedding, and two points that are poorly connected will tend to be mapped to distant points in the embedding. There are other interpretations in terms of Markov probabilities.
This is very important, because this gives rise to all sorts of dimensionality reduction schemes. In applications, that means you can take large high-dimensional datasets (where computations or machine learning are difficult), and meaningfully embed them into lower-dimensional Euclidean spaces $\mathbb R^n$ where you can run more efficient clustering algorithms and ML.
If this  seems a bit too  abstract, consider the following example: If you take a simple "loop" graph: $n$ vertices $v_1,...,v_{n-1}$, where $v_i$ is connected to $v_{i-1}$ and $v_{i+1}$ (and $v_1$ is connected to $v_n$ and $v_2$). Then your Laplacian's eigenvectors will be the Fourier basis of $\cos$ and $\sin$ at different frequencies (eigenvalues). The first 2 eigenvectors ($\cos$ and $\sin$, ignoring the  constant function) will map your graph onto a circle, and each vertex can be parametrized by  the angle on that circle. Note that we recovered that your graph was of "dimension 1".
Thee image below represents a 1d manifold (the wiggly curve in $\mathbb R^3$) and how it gets mapped into a circle in the plane using the first 2 eigenvectors.

