  Given a finite transitive permutation group $G\leq
\operatorname{Sym}(\Omega)$, with $|\Omega|\geq 2$, the derangement graph
$\Gamma_G$ of $G$ is the Cayley graph
$\operatorname{Cay}(G,\operatorname{Der}(G))$, where $\operatorname{Der}(G)$ is
the set of all derangements of $G$. Meagher et al. [On triangles in derangement
graphs, {\it J. Combin. Theory Ser. A}, 180:105390, 2021] recently proved that
$\operatorname{Sym}(2)$ acting on $\{1,2\}$ is the only transitive group whose
derangement graph is bipartite and any transitive group of degree at least
three has a triangle in its derangement graph. They also showed that there
exist transitive groups whose derangement graphs are complete multipartite.
  This paper gives two new families of transitive groups with complete
multipartite derangement graphs. In addition, we prove that if $p$ is an odd
prime and $G$ is a transitive group of degree $2p$, then the independence
number of $\Gamma_{G}$ is at most twice the size of a point-stabilizer of $G$.
