TITLE: Intuitively, what does a graph Laplacian represent?
QUESTION [58 upvotes]: Recently I saw an MO post  Algebraic graph invariant $\mu(G)$ which links Four-Color-Theorem with Schrödinger operators: further topological characterizations of graphs? that got me interested. It is about a graph parameter that is derived from the Laplacian of a graph. Its origins are in spectral operator theory, but it is quite strong in characterizing important properties of graphs. So I was quite fascinated by the link it creates between different branches of mathematics.
I went through other posts on MO that discuss this topic as well, and in the meantime I read a few linked articles that work with the graph Laplacian. I understand that they view an (undirected) graph as a metric graph embedded in a surface, and the metric on the graph is approximated by Riemannian metrics which give the edge distance along the edges, and which is close to zero everywhere else on the surface. The eigenvalues of the surface Laplacian approximate the eigenvalues of the graph Laplacian, and a lot of surprisingly useful conclusions follow, about connectivity and embeddability of the graph, and even about minor-monotonicity.
I have gained a technical understanding of what is happening and how these eigenvalues (and their multiplicity) are determined, using the graph Laplacian. I also have a basic understanding of the role of a Laplacian in differential geometry, like the Laplacian of a function $f$ at a point $x$ measures by how much the average value of $f$ over small spheres around $x$ deviates from $f(x)$, or I think of it to represent the flux density of the gradient flow of $f$.

But I am failing to gain or develop such an intuition for the graph Laplacian. Conceptually or intuitively, what does a graph Laplacian represent? I am trying to understand, how can it be so powerful when applied to graphs? (I am aware that the graph Laplacian can be defined using the graph adjacency matrix, but I was unable to link this with my differential geometry intuition)

REPLY [17 votes]: This is just a long comment, adding to the excellent answers above.
There is a great article from László Lovász "Discrete and Continuous:
Two sides of the same?", written around 2000 (https://web.cs.elte.hu/~lovasz/telaviv.pdf) which might be of interest to you. In chapter 5 of this article, Lovász covers the graph Laplacian. He explains the relation to random walks on graphs and also the link to the Colin de Vérdière graph invariant which sparked your interest (your link in the OP).
In your OP, you are asking how can the graph Laplacian be so powerful when applied to graphs? I think two quotes from this article could be of special interest to you, because quote (1) relates to the "power" and quote (2) relates to where the "limitations" were in applying the graph Laplacian.
About the "power":

Quote (1)
"The Laplacian makes
sense in graph theory, and in fact it is a basic tool. Moreover, the study of the discrete and
continuous versions interact in a variety of ways, so that the use of one or the other is almost
a matter of convenience in some cases. (...) Colin de Verdière’s invariant created much interest among graph theorists, because of its
surprisingly nice graph-theoretic properties. (...) Moreover, planarity of graphs can be
characterized by this invariant: $\mu(G) \leq 3$ if and only if G is planar.
Colin de Verdière’s original proof of the “if” part of this fact was most unusual in graph
theory: basically, reversing the above procedure, he showed how to reconstruct a sphere and a
positive elliptic partial differential operator $P$ on it so that $\mu(G)$ is bounded by the dimension
of the null space of $P$, and then invoked a theorem of Cheng (...) asserting that this dimension is at most $3$.

About the "limitations":

Quote (2)
"Later Van der Holst (...) found a combinatorial proof of this fact [$\mu(G) \leq 3$ if and only if G is planar]. While this may seem as a step backward (after all, it eliminated the necessity of the only application of partial differential equations in graph theory I know of), it did open up the possibility of characterizing the next case. Verifying a conjecture of Robertson, Seymour, and Thomas, it was shown by Lovász and Schrijver (...) that $\mu(G) \leq 4$ if and only if G is linklessly embedable in $\mathbb R^3$."