If a graph G has e edges, v vertices, and p components, the rank of G, written ρ(G), is defined to be v – p
The nullity of G, written μ(G), is e – v + p Thus ρ(G) + μ(G) = e
With reference to a directed graph, a weakly connected graph is one in which the direction of each edge must be removed before the graph can be connected in the manner described above. If however there is a directed path between each pair of vertices u and v and another directed path from v back to u, the directed graph is strongly connected.
More formally, let G be a directed graph with vertices V and edges E. The set V can be partitioned into equivalence classes V1, V2,… under the relation that vertices u and v are equivalent iff there is a path from u to v and another from v to u. Let E1, E2,… be the sets of edges connecting vertices within V1, V2,… Then each of the graphs Gi with vertices Vi and edges Ei is a strongly connected component of G. A strongly connected graph has precisely one strongly connected component.
The process of replacing each of the strongly connected components of a directed graph by a single vertex is known as condensation.
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