Circuit Rank

The circuit rank (also called the cycle rank or cyclomatic number) of a connected graph tells you how many edges must be removed to eliminate all cycles and produce a spanning tree.

Let G be a connected graph with n vertices and m edges. A spanning tree of G contains exactly (n − 1) edges. Therefore, the number of edges you need to delete from G to get a spanning tree is −

Circuit Rank = m − (n − 1)

This formula works because a spanning tree must have exactly n − 1 edges. The remaining m − (n − 1) edges are the "extra" edges that create cycles in the graph.

Example 1: Direct Calculation

Take a look at the following graph ?

For this graph, m = 7 edges and n = 5 vertices −

Circuit Rank = m - (n - 1)
             = 7 - (5 - 1)
             = 7 - 4
             = 3

This means 3 edges must be removed to eliminate all cycles and obtain a spanning tree (which would have 4 edges).

Example 2: Using Sum of Degrees

Let G be a connected graph with 6 vertices and the degree of each vertex is 3. Find the circuit rank of G.

First, find the number of edges using the sum of degrees theorem −

Sum of degrees = 2 × |E|

6 × 3 = 2 × |E|

18 = 2 × |E|

|E| = 9

Now calculate the circuit rank −

Circuit Rank = |E| - (|V| - 1)
             = 9 - (6 - 1)
             = 9 - 5
             = 4

The circuit rank is 4, meaning 4 edges must be removed to make the graph cycle-free.

Conclusion

The circuit rank of a connected graph measures the number of independent cycles in the graph. It equals m − (n − 1), where m is the number of edges and n is the number of vertices. A tree (which has no cycles) always has a circuit rank of 0.