  Let $G$ be a simple graph of order $n$. The double vertex graph $F_2(G)$ of
$G$ is the graph whose vertices are the $2$-subsets of $V(G)$, where two
vertices are adjacent in $F_2(G)$ if their symmetric difference is a pair of
adjacent vertices in $G$. A generalization of this graph is the complete double
vertex graph $M_2(G)$ of $G$, defined as the graph whose vertices are the
$2$-multisubsets of $V(G)$, and two of such vertices are adjacent in $M_2(G)$
if their symmetric difference (as multisets) is a pair of adjacent vertices in
$G$. In this paper we exhibit an infinite family of graphs (containing
Hamiltonian and non-Hamiltonian graphs) for which $F_2(G)$ and $M_2(G)$ are
Hamiltonian. This family of graphs is the set of join graphs $G=G_1 + G_2$,
where $G_1$ and $G_2$ are of order $m\geq 1$ and $n\geq 2$, respectively, and
$G_2$ has a Hamiltonian path. For this family of graphs, we show that if $m\leq
2n$ then $F_2(G)$ is Hamiltonian, and if $m\leq 2(n-1)$ then $M_2(G)$ is
Hamiltonian.
