This is an introduction to graph theory, from a geometric viewpoint. A finite graph $X$ is described by its adjacency matrix $d\in M_N(0,1)$, which can be thought of as a kind of discrete Laplacian, and we first discuss the basics of graph theory, by using $d$ and linear algebra tools. Then we discuss the computation of the classical and quantum symmetry groups $G(X)\subset G^+(X)$, which must leave invariant the eigenspaces of $d$. Finally, we discuss similar questions for the quantum graphs, with these being again described by certain matrices $d\in M_N(\mathbb C)$, but in a more twisted way.