Vertex-Minimal Graphs and Fixing Sets for Modular p-groups

Abstract

Let G be a finite group. Define α(G) as the minimum number of vertices among all graphs Γ such that Aut Γ ∼= G. For any p prime, all p-groups of order p n having cyclic subgroups of order p n − 1 have been completely classified. Here, we consider one family of groups called modular p-groups, denoted by Modn(p), for an odd prime p and n ≥ 3. We compute the order of vertexminimal graphs with Modn(p)-symmetry. The fixing number of a graph Γ is defined as the smallest number of vertices in V (Γ) that, when fixed, makes Aut Γ trivial. For a finite group G, the fixing set is defined as the set of all fixing numbers of graphs having automorphism groups isomorphic to G. We show that any graph Γ whose automorphism group is a modular p-group has the fixing number 1. As a result, modular p-group’s fixing set becomes {1}.