The circuit rank of a complete graph with n=4 (6 edges)is 3. The circuit rank of a complete graph with n=5 (10 edges)is 5. I think that the circuit rank of a complete graph with n=6 (15 edges) is 10? I think that the circuit rank of a complete graph with n=7 (21 edges) is 15?
I don't see the pattern.
The circuit rank/nullity '$\mu$' of a graph is $\mu=e-n+k$, where $e$ is the number of edges, $n$ the number of vertices and $k$ the number of maximally connected subgraphs/components. In the case of complete graphs, $e=\binom n2$ and $k=1$. Thus,$$\mu=\binom n2-(n-1)=\frac{n(n-1)}2-(n-1)=\frac{(n-1)(n-2)}2$$Also note that the circuit rank of complete graph of $5$ vertices i.e. $K_5$ is $6$, not $5$.