Constructing physically intuitive graph invariants

By a graph, a mathematician generally means a collection of points (vertices) and a list indicating how they are connected (a collection of edges). The study of graphs and their properties is a huge industry, with applications from the completely abstract (e.g. classification of algebras) to the very applied (e.g. network routing). The simplest type of graph, and the only type which I’ll consider here, is one that has either zero or one (no multiple) undirected connections between any two vertices. Given two graphs, such as in Fig. 1., one of the simplest questions to ask is whether they are actually the same graph; this is known as the graph isomorphism problem. If the two graphs are different, the question is often simple to decide. For example, the two graphs may have different numbers of vertices, although these two do not. If they have the same number of vertices they may have a different number of edges, although again these two do not. If they have the same number of vertices and edges, it may be, as in Fig.1., that only one of the graphs has a vertex which is connected to exactly four other vertices, indicating clearly the graphs cannot be isomorphic. More generally, we can list the degrees of each graph’s vertices and check if the lists are identical. (Note that we are only interested in the underlying connectivity of the graph, and so the distances between vertices are not important; the same graph can be drawn many different ways.) Once we have performed these few simple checks, which I should emphasize are capable only of telling us whether the graphs are different, things get a little trickier. To be convinced that two graphs are the same we need to find a map of the vertices of graph 2 to those of graph 1. That is, we try and find a relabelling of the second graph’s vertices such that it is now manifestly clear that it is the same as the first graph. The problem is that the number of ways we can re-label the N vertices of a graph is N ! – the number of permutations of N items. This amounts to a very large number of possible relabellings, and for modest values of N searching through them all becomes computationally infeasible. (Of course if someone magically hands us the correct relabelling it is very easy to check that the two graphs are the same!)