Is there any relation between Global minimum cut problem and Maximal independent set?
Minimum cut can be computed efficiently (you haven't explained what global minimum cut is, but presumably it reduces to minimum $s$-$t$ cut). Minimum vertex cover (probably) cannot. So any connection would be rather indirect. Minimum cut is related to another quantity, maximum flow, which is defined more generally for weighted directed graphs. Both quantities are equal due to the fundamental max-flow min-cut theorem (sometimes known as the min-cut max-flow theorem).