Extremal Problem for Graphs with Modular p-group Symmetry

Abstract

For a finite group 𝐺, define 𝛼(𝐺) as the minimum number of vertices among all graphs Γ such that Aut Γ ∼= 𝐺 [1]. For any 𝑝 prime, all 𝑝-groups of order 𝑝𝑛 having cyclic subgroups of order 𝑝𝑛−1 have been completely classified. Several authors have already investigated some of these families of groups in order to find vertex-minimal graphs [2]. Here we consider a family of groups called modular 𝑝-groups, for an odd prime 𝑝 and 𝑛 ≥ 3. A modular 𝑝-group is defined as Mod𝑛(𝑝) = ⟨𝑎𝑝𝑛−1 = 1, 𝑏𝑝 = 1, 𝑏𝑎 = 𝑎𝑝𝑛−2+1𝑏⟩. We compute the order of vertex-minimal 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. This concept has been extended to finite groups by Gibbons and Laison [3]. 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, the modular 𝑝-group’s fixing set becomes {1}. Theorem 1. Let 𝑝 be odd prime and 𝑛 ≥ 3 be an integer. Then 𝛼(Mod𝑛(𝑝)) = 𝑝𝑛. Using the theory of transitive permutation groups, we discussed the cycle decomposition of the generators of modular 𝑝-group and then evaluated the fixing set. Theorem 2. Let 𝑝 be odd prime and 𝑛 ≥ 3 be an integer. Then fix(Mod𝑛(𝑝)) = {1}. Example. Let 𝐺 = Mod3(3) and 𝑆 = {𝛽, 𝛾, 𝛽8, 𝛾2, 𝛽𝛾, 𝛾2𝛽8} ⊂ 𝐺. The Cayley graph 𝐶𝑎𝑦(𝐺, 𝑆) has automorphism group isomorphic to G with 27 vertices and the edge set 𝐸(𝐶𝑎𝑦(𝐺, 𝑆)), that comprises the edge orbits 𝑂{𝛽, 𝛽2}, 𝑂{𝛽, 𝛽𝛾}, 𝑂{𝛽, 𝛽2𝛾}, 𝑂{𝛽, 𝛽𝛾2}, and 𝑂{𝛽, 𝛽3𝛾2}.