The most common way to store a graph in Python is an adjacency list: a dictionary mapping each node to a list of its neighbours.

roads = {
    "Ashford": ["Canterbury", "Dover", "Maidstone"],
    "Canterbury": ["Ashford", "Dover", "Margate"],
    "Dover": ["Ashford", "Canterbury"],
    "Maidstone": ["Ashford"],
    "Margate": ["Canterbury"],
}

print(roads["Canterbury"])
print("Dover" in roads["Ashford"])
print(len(roads), "towns,", sum(len(v) for v in roads.values()) // 2, "roads")

For an undirected graph, each edge appears twice: Ashford lists Dover, and Dover lists Ashford. Building one from a list of edges is a good use of defaultdict:

from collections import defaultdict

edges = [("A", "B"), ("A", "C"), ("B", "D")]
graph = defaultdict(list)
for a, b in edges:
    graph[a].append(b)
    graph[b].append(a)
print(dict(graph))