TITLE: Uses of the incidence matrix of a graph
QUESTION [8 upvotes]: The incidence matrix of a graph is a way to represent the graph. Why go through the trouble of creating this representation of a graph? In other words what are the applications of the incidence matrix or some interesting properties it reveals about its graph?

REPLY [5 votes]: Another example is the beautiful Matrix Tree Theorem, which says that the number of spanning trees of a graph is equal to a minor of the Laplacian of the graph, which is a matrix closely related to the incidence matrix.