  The max-cut problem is a classical graph theory problem which is NP-complete.
The best polynomial time approximation scheme relies on \emph{semidefinite
programming} (SDP). We study the conditions under which graphs of certain
classes have rank~1 solutions to the max-cut SDP. We apply these findings to
look at how solutions to the max-cut SDP behave under simple combinatorial
constructions. Our results determine when solutions to the max-cut SDP for
cycle graphs are rank~1. We find the solutions to the max-cut SDP of the
vertex~sum of two graphs. We then characterize the SDP solutions upon joining
two triangle graphs by an edge~sum.
