Graphs and their symmetries

TL;DR

This paper introduces graph theory from a geometric and analytic perspective, focusing on classical and quantum symmetries of graphs via their adjacency matrices and eigenspaces.

Contribution

It provides a detailed analysis of classical and quantum symmetry groups of graphs, highlighting the differences and methods to compute them using linear algebra.

Findings

Quantum symmetry groups are generally larger than classical ones.

Eigenvalue invariance characterizes symmetry groups.

Linear algebra tools facilitate symmetry group computation.

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 $G(X)$.