How do I get the vertices on the shortest path using igraph?

The function you need is get_shortest_paths I believe. See https://igraph.org/python/doc/igraph.GraphBase-class.html#get_shortest_paths

You need to call it individually for each source vertex, and it will give you only a single (arbitrary) shortest path for each pair of nodes. If you need all shortest paths, then see get_all_shortest_paths: https://igraph.org/python/doc/igraph.GraphBase-class.html#get_all_shortest_paths

I do this

from igraph import *
g = Graph([(0,1), (0,2), (2,3), (3,4), (4,2), (2,5), (5,0), (6,3), (5,6)])
g.vs["name"] = ["Alice", "Bob", "Claire", "Dennis", "Esther", "Frank", "George"]
#You could create Vertexes like g.add_vertex(name="Bill") 
path=g.get_shortest_paths("Alice",to="Frank",mode=OUT,output='vpath')
for n in path[0]:
    print("{}".format(g.vs[n]['name']))

Hope this helps

This is the way to find shortest path for weighted directed graph (DAG). So this what I figured out:

import igraph
from igraph import *
g = Graph(directed=True)
g.add_vertices(3)
g.vs["name"]=["GO:1234567","GO:6789056","GO:5674321"]
g.es["weight"]=1
g['GO:1234567','GO:6789056']=1
g['GO:6789056','GO:5674321']=5
weight=g.es["weight"]
print weight
print g.degree(mode="in") 
print g.shortest_paths_dijkstra(source="GO:1234567", target="GO:5674321", 
    weights=weight, mode=OUT)