The matrix $A_{n}$ which is the adjacency matrix of a directed path on $n$ vertices seems to work pretty well.

Some values I've computed for it:

$c(A_{5})=4$

$c(A_{6})=8$

$c(A_{7})=10$

$c(A_{8})=16$

Perhaps this is related to the OEIS sequence A005232 but computing $c(A_{9})$ was too much for my computer...

The matrix $A_{n}$ which is the adjacency matrix of a directed path on $n$ vertices seems to work pretty well.

For example:

$A=\begin{bmatrix}0&1&0&0&0\\\\ 0&0&1&0&0\\\\ 0&0&0&1&0\\\\ 0&0&0&0&1\\\\ 0&0&0&0&0\\\\ \end{bmatrix}$.

Some values I've computed for it:

$c(A_{5})=4$

$c(A_{6})=8$

$c(A_{7})=10$

$c(A_{8})=16$

Perhaps this is related to the OEIS sequence A005232 but computing $c(A_{9})$ was too much for my computer...