Clique-sums have a close connection with treewidth: If two graphs have treewidth at most k, so does their k-clique-sum.
There is also a close connection between clique-sums and graph connectivity: if a graph is not (k + 1)-vertex-connected (so that there exists a set of k vertices the removal of which disconnects the graph) then it may be represented as a k-clique-sum of smaller graphs.