Graphs and their symmetries

Abstract

This is an introduction to graph theory, from a geometric and analytic viewpoint. A finite graph $X$ is described by its adjacency matrix $d\in M_N(0,1)$, which can be thought of as being a kind of discrete Laplacian, and we first discuss the basics of graph theory, by using $d$, and various 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 $d$, with the quantum symmetry group $G^+(X)$ being in general bigger than the classical symmetry group $G(X)$.