Count all possible walks from a source to a destination with exactly k edges based on http://www.geeksforgeeks.org/count-possible-paths-source-destination-exactly-k-edges/

count_walks.py

def count_walks(graph, u, v, k):
    if(k == 0 and u == v): return 1
    if(k == 1 and graph[u][v]): return 1
    if(k <= 0): return 0

    count = 0

    for i in range(0, 5):
        if(graph[u][i]):
            count += count_walks(graph, i, v, k-1)
    return count


graph = [[0, 1, 0, 1, 0],
[0, 0, 1, 0, 0],
[0, 0, 0, 1, 1],
[0, 0, 1, 0, 1],
[0, 1, 0, 0, 0]]

k = 4
u = 0
v = 2
print(count_walks(graph, u, v, k))

k = 3
u = 2
v = 2
if(u == v):
    k += 1
print(count_walks(graph, u, v, k))