Topics in Graph Automorphisms

The symmetry of a graph is measured by its automorphism group: the set of permutations of the vertices so that all edges and non-edges are preserved. There are natural questions which arise when considering the automorphism group and there are several interesting results in this area that are not well-known. This talk presents some of these results and maybe even proves one or two. First, we will prove that almost all graphs have trivial automorphism group. Second, we will briefly discuss the relation of the graph automorphism and group intersection problems. Then, we will discuss Babai’s constrcution that any finite group with n elements can be represented by a graph on 2n vertices (other than three exceptions). Finally, we will mention there exists a subgroup of Sn that is the automorphism group of no graph of size less than 1 2 ( n 1 2 n ). 1 Almost all graphs are rigid Before we can begin the proof of this fact, recall the Chernoff-Hoeffding bounds. Theorem 1.1 ([DP09]). Let X = ∑n i=1 Xi be a sum of identically distributed independent random variables Xi where Pr(Xi = 1) = p, Pr(Xi = 0) = q = 1− p. Then, we have the following relative Chernoff-Hoeffding bound for all ε > 0: Pr[X < (1− ε)np] ≤ e−npε/2, Pr[X > (1 + ε)np] ≤ e−npε/2 1.1 Properties of G(n, p) Lemma 1.2. Let ε be a function on n with ε(n) > 0. Then, the probability that G, distributed as G(n + 1, p), has all vertices of degree deg v ∈ ((1− ε)np, (1 + ε)np) is at least 1− 2(n + 1)e− npε2 2 . Proof. Let Xi, j be the indicator variable for the edge {i, j} appearing in G(n + 1, p) (1 ≤ i < j ≤ n + 1). The expected value is p. By linearity of expectation,