and only if it is non-abelian and every commutator is central.

A *group isomorphism* is a bijective map $\varphi\colon G\to H$ that preserves group multiplication. We collect all isomorphisms between $G$ and $H$ in a (possibly empty) set $\operatorname{Iso}(G,H)$ and set $\operatorname{Aut}(G):=\operatorname{Iso}(G,G)$. We write $\operatorname{Sub}(G)$ for the set of all subgroups of $G$.

We assume the term ’group’ to mean ’finite group’ and whenever we include infinite groups we do so explicitly.

#### Graphs.

All graphs will be finite simple undirected graphs and referred to with greek characters, primarily $\Gamma$, subject to suitable subscripts. That is, a graph $\Gamma$ consists of a finite set of vertices $V(\Gamma)$ and a set of edges $E(\Gamma)\subseteq{{V(\Gamma)}\choose{2}}:=\{M\subseteq V(\Gamma)\mid |M|=2 \}$. The *complement* of $\Gamma$ will always be the *simple* complement graph, namely $\operatorname{co}(\Gamma):=\left(V(\Gamma),{{V}\choose{2}}-E(\Gamma)\right)$. The set of *neighbors* of $v\in V(\Gamma)$ is $N(v):=\{w\in V(\Gamma)\mid \{v,w\}\in E\}$ and $N[v] := N(v)\cup\{v\}$ is the *closed neighborhood* of $v$. The *degree* of $v$ is $d(v):=|N(v)|$. A graph is *$d$-regular* if every vertex has degree $d$. For a set of vertices $M\subseteq V(\Gamma)$, the *induced subgraph* is $\Gamma[M]:=\left(M,E(\Gamma)\cap
{{M}\choose{2}}\right)$.

An *isomorphism* of graphs is a bijective map $\varphi\colon V(\Gamma_1)\to V(\Gamma_2)$ that simultaneously preserves edges and non-edges. The set of 