  This paper considers the edge-connectivity and restricted edge-connectivity
of replacement product graphs, gives some bounds on edge-connectivity and
restricted edge-connectivity of replacement product graphs and determines the
exact values for some special graphs. In particular, the authors further
confirm that under certain conditions, the replacement product of two Cayley
graphs is also a Cayley graph, and give a necessary and sufficient condition
for such Cayley graphs to have maximum restricted edge-connectivity. Based on
these results, the authors construct a Cayley graph with degree $d$ whose
restricted edge-connectivity is equal to $d+s$ for given odd integer $d$ and
integer $s$ with $d \geqslant 5$ and $1\leqslant s\leqslant d-3$, which answers
a problem proposed ten years ago.
