Examples of theorems in graph theory I'm looking for examples of uses of the incidence matrix in graphs for my class (apart from proving that for a graph $G=(V,E)$ we have $2|E|=\underset{v\in V}{\sum}d(v)$).
Can you think of anything interesting?
Thanks in advance for any help.
A classic example is the Friendship theorem of Erdos and Renyi.
There is also the classical result that the powers of the adjacency matrix describe the number of paths of a given length between two vertices.