Time complexity to check if there is an edge between two nodes in an adjacency list
In an adjacency list each vertex $u \in V$ is associated with a list of adjacent vertices.
Given a graph $G=(V,E)$, in order to check if the edge $(u,v) \in E$ you need to check whether $v \in \text{adjacent[u]}$.
A node can have at most $O(|V|)$ neighbors, from here the complexity follows.