  We derive a correspondence between the eigenvalues of the adjacency matrix
$A$ and the signless Laplacian matrix $Q$ of a graph $G$ when $G$ is
$(d_1,d_2)$-biregular by using the relation $A^2=(Q-d_1I)(Q-d_2I)$. This
motivates asking when it is possible to have $X^r=f(Y)$ for $f$ a polynomial,
$r>0$, and $X,\ Y$ matrices associated to a graph $G$. It turns out that,
essentially, this can only happen if $G$ is either regular or biregular.
