  A path in a vertex-colored graph is a {\it vertex-proper path} if any two
internal adjacent vertices differ in color. A vertex-colored graph is {\it
proper vertex $k$-connected} if any two vertices of the graph are connected by
$k$ disjoint vertex-proper paths of the graph. For a $k$-connected graph $G$,
the {\it proper vertex $k$-connection number} of $G$, denoted by $pvc_{k}(G)$,
is defined as the smallest number of colors required to make $G$ proper vertex
$k$-connected. A vertex-colored graph is {\it strong proper vertex-connected},
if for any two vertices $u,v$ of the graph, there exists a vertex-proper
$u$-$v$ geodesic. For a connected graph $G$, the {\it strong proper
vertex-connection number} of $G$, denoted by $spvc(G)$, is the smallest number
of colors required to make $G$ strong proper vertex-connected. In this paper,
we study the proper vertex $k$-connection number and the strong proper
vertex-connection number on the join of two graphs, the Cartesian,
lexicographic, strong and direct product, and present exact values or upper
bounds for these operations of graphs.
