On finite groups whose power graphs satisfy certain connectivity conditions

TL;DR

The paper investigates connectivity properties of power graphs $\mathcal{P}(G)$ for finite groups, focusing on cyclic, dihedral, and dicyclic types. It provides a complete characterization of when $\mathcal{P}(G)$ is cyclically separable and when the vertex connectivity $\kappa(\mathcal{P}(G))$ equals the cyclic vertex connectivity $c\kappa(\mathcal{P}(G))$ for cyclic groups, with precise number-theoretic conditions on $n$ in $G \cong C_n$. For noncyclic groups, in particular $D_{2n}$ and $Q_{4n}$, it shows that cyclic separability aligns with the corresponding cyclic-subgroup graphs but the two connectivities differ ($\kappa(\mathcal{P}(D_{2n})) = 1$ and $\kappa(\mathcal{P}(Q_{4n})) = 2$, while $c\kappa$ exceeds these values), thereby ruling out equality in these cases. The results extend prior work on $p$-groups and clarify the relationship between group structure and the connectivity of associated power graphs, offering a clear dichotomy between cyclic and noncyclic groups. The methods combine structural analysis of order classes in $C_n$ with careful minimal cutset arguments to derive sharp, order-dependent criteria.

Abstract

Consider a graph $Γ$. A set $ S $ of vertices in $Γ$ is called a {cyclic vertex cutset} of $Γ$ if $Γ- S$ is disconnected and has at least two components containing cycles. If $Γ$ has a cyclic vertex cutset, then it is said to be {cyclically separable}. The {cyclic vertex connectivity} is the minimum cardinality of a cyclic vertex cutset of $Γ$. The power graph $\mathcal{P}(G)$ of a group $G$ is the undirected simple graph with vertex set $G$ and two distinct vertices are adjacent if one of them is a positive power of the other. If $G$ is a cyclic, dihedral, or dicyclic group, we determine the order of $G$ such that $\mathcal{P}(G)$ is cyclically separable. Then we characterize the equality of vertex connectivity and cyclic vertex connectivity of $\mathcal{P}(G)$ in terms of the order of $G$.

Figures (3)

• Figure 1: $\mathcal{P}'(C_n)$
• Figure 2: $\mathcal{P}'(C_n)$
• Figure 3: $\mathcal{P}'(C_n)$

Theorems & Definitions (26)

• Lemma 1.1
• Lemma 1.2: panda2018a
• Lemma 1.3: panda2018a
• Lemma 1.4: panda2018a
• Lemma 2.1
• Theorem 2.2: panda2024
• Theorem 2.3: panda2024
• Theorem 2.4
• proof
• Lemma 2.5