Betti number
Simplicial homology allows the computation of the rank of "H", which is equal to "h"1("Y"), the first Betti number of the covering space, the number of independent cycles. For "G" free of rank "n", the graph "X" has "n" edges and 1 vertex; assuming "H" has finite index ["G" : "H"] = "e", the covering graph "Y" has "en" edges and "e" vertices. The first Betti number of a graph is equal to the number of edges, minus the number of vertices, plus the number of connected components; hence the rank of "H" is: