Smallest graphs with given automorphism group

Abstract

For a finite group G, denote by α(G) the minimum number of vertices of any graph Γ having Aut(Γ) ∼ = G. In this paper, we prove that α(G) ≤ |G|, with specified exceptions. The exceptions include four infinite families of groups, and 17 other small groups.

Smallest graphs with given automorphism group

“…Assume now that p r 1 1 p r 2 2 < 9; thus, p r 1 1 = 2 and 2 ≤ p r 2 2 ≤ 4. If p r 3 3 < 16 then using Table 3 we verify that the claim in Proposition 3.1 holds for G. If p r 3 3 ≥ 16 instead, then (1) and (3) together with Proposition 3.3 imply that…”

“…The proof is more technical but similar to others in this section. For a detailed justification, we refer the reader to the arXiv version of this article [3]. Practically, φ interchanges two pairs of vertices in the orbit of size 6, if it exists, one pair in the orbit of size 3, if both an orbit of size 3 and 6 exist, and one pair in every orbit of size 4.…”

“…The arguments in [21,Proposition 3.7] extend to the case that contains an additional orbit of size or up to three additional orbits of size 1. For details on the extension we refer the reader to the arXiv version of this article [3].…”