Method 1 - The Adjacency Matrix

I'll show you the adjacency matrix. You store a $2$D array $M$ where M[i][j] = 1 if there is an edge from vertex $i$ to vertex $j$, and $0$ otherwise. In an undirected graph the matrix is symmetric because $i$ connects to $j$ and $j$ connects to $i$.

Checking whether edge $(u, v)$ exists takes $O(1)$ time because you read M[u][v]. The cost is space. You store $V^2$ entries, so it uses $O(V^2)$ space even when the graph is sparse. If you are memory bound, you will feel this immediately.