Is there anything known about the least-reliable simple graphs, among all graphs $G$ with edge-connectivity $c \leq n$?
Here, least-reliable means the lowest all-terminal reliability, i.e. the probability that the graph $G$ remains connected, when edges fail independently with probability $p$
If you don't restrict to simple graphs, then it is known that the cycle graph with $c/2$ edges between successive vertices is the least simple for a fixed value of $c$. But this graph is far from simple.