Assume $G$ is a simple $k$-regular graph of order $n$ with adjacency matrix $A$ which is non-singular.


Does anyone know some lower bounds for $\vert \det (A) \vert$ with respect to $n$, $k$ or both?


Thanks in advance.



Lower bound for $\vert \det A \vert $ for the adjacency matrix of regular graphs
$k$ is a lower bound, since $k$ is an eigenvalue and the product of other eigenvalues is integer (they are roots of the monic polynomial with integer coefficients). I doubt that it may be improved without further assumptions.


