Spectral Graph Theory Lecture 24 Strongly Regular Graphs , part 2

In this lecture, I will present three results related to Strongly Regular Graphs. 1. An algorithm for testing isomorphism of SRGs that runs in time 2 O(√ n log n). 2. A proof that f and g, the dimensions of the eigenspaces, are both at least √ n. The problem of testing isomorphism of graphs is often reduced to the problem of giving each vertex in a graph a unique name. If we have a way of doing this that does not depend upon the initial ordering of the vertices, then we can use it to test graph isomorphism: find the unique names of vertices in both graphs, and then see if it provides an isomorphism. For example, consider the graph below. We could begin by labeling every vertex by its degree. 1 2 3 2 1 3 The degrees distinguish between many nodes, but not all of them. We may refine this labeling by appending the labels of every neighbor of a node.