Maximum Adjacency Ordering

Start with an arbitrary vertex in the ordering. Repeatedly add the vertex with the maximum total weight of edges to already-added vertices. This is a greedy selection process. The last two vertices added are $t$ (last) and $s$ (second-to-last). Compute the minimum $s$-$t$ cut (which is the sum of weights of edges from $t$ to already-added vertices).

This gives you a cut candidate. This cut is a candidate for the global minimum cut. Track the smallest cut found across all iterations of the algorithm.