Think about the edges. Since the center connects to everyone, the center node must be part of every single edge in the list.
Look at any two edges. They share exactly one node in common: the center. So you can find the center by looking at the first two edges and finding which node appears in both.
center := intersection of edge[0] and edge[1]
This runs in time regardless of graph size. You do not need to build any data structure or count degrees. Two edges are enough to identify the unique center node.
Space complexity is for the data structures used.