  A graph $G$ of order $n>2$ is pancyclic if $G$ contains a cycle of length $l$
for each integer $l$ with $3 \leq l \leq n $ and it is called vertex-pancyclic
if every vertex is contained in a cycle of length $l$ for every $3 \leq l \leq
n $. A graph $G$ of order $n > 2$ is Hamilton-connected if for any pair of
distinct vertices $u$ and $v$, there is a Hamilton $u$-$v$ path, namely, there
is a $u$-$v$ path of length $n-1$. The graph $ B(n)$ is a graph with the vertex
set $V=\{v \ | \ v \subset [n] , | v | \in \{ 1,2 \} \} $ and the edge set $ E=
\{ \{ v , w \} \ | \ v , w \in V , v \subset w $ or $ w \subset v \}$, where
$[n]=\{1,2,...,n\}$. We denote by $L(n)$ the line graph of $B(n)$, that is,
$L(n)=L(B(n))$. In this paper, we show that the graph $L(n)$ is
vertex-pancyclic and Hamilton-connected whenever $n\geq 6$.
