Python 函数：检查邻接矩阵中的连通性

Question

I have an adjacency matrix D below. How do I write up a python function that returns True if all the vertices in the matrix are connected or False if not?

 D = [['a', 'c', 'g', 'w', 'Q', 'f', 'Z', 't', 'R'], [0, 1, 2, 1, 9, 0, 0, 0, 0], [1, 0, 3, 4, 0, 0, 0, 0, 0], [2, 3, 0, 15, 2, 0, 0, 0, 0], [1, 4, 15, 0, 7, 0, 0, 0, 0], [9, 0, 2, 7, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 9, 0], [0, 0, 0, 0, 0, 2, 0, 0, 20], [0, 0, 0, 0, 0, 9, 0, 0, 0], [0, 0, 0, 0, 0, 0, 20, 0, 0]] def connectivity(adjMatrix): connected = True while connected == True: # some algorithm that checks that each vertex can be connected to any other vertex # if connected -> remains True # if not connected -> False return connected print(connectivity(D))

Answer 1

You can use DFS or depth first search. You only need to run on one vertex, because if one vertex is connected to all the nodes, that means there is full connectivity within the graph.

Here is pseudocode for a recursively implemented DFS (using the call stack):

def DFS(vertex, adj, vis):
    # adj is the adjacency matrix and vis is the visited nodes so far
    set vertex as visited # example if vis is list: vis[vertex] = True
    for vert in adj[vertex]:
        if vert is not visited:
            DFS(vertex, adj, vis)
    return whether or not all vertices are visited # this only needs to happen 
                                                    # for the first call

This algorithm will have a runtime of O(n) with a space complexity of O(n) (for the vis array).

Answer 2

Since this is the answer that turns up for searches of "check connectivity of graph's adjacency matrix", let's actually answer that instead of leaving it at "this is a well understood topic".

Just use NetworkX's is_connected function .

Suppose your adjacency matrix is already in numpy format:

# An adjacency matrix is a square, binary matrix.
G = nx.from_numpy_matrix(adj_matrix)
if nx.is_connected(G):
    pass  # We're done! That easy.

If you were moreso interested in connected components, as opposed to the whole graph, read here .

If you needed to input your adjacency matrix from a different format, try here .

Also of interest for those who use graph theory: the algebraic connectivity of a graph .