  Determinantal ideals of graphs generalize, among others, the spectrum and the
Smith normal form (SNF) of integer matrices associated to graphs. In this work
we investigate the relationship of the spectrum and the SNF with the
determinantal ideals. We show that an eigenvalue divides the $k$-th invariant
factor of its SNF if the eigenvalue belongs to a variety of the $k$-th
univariate integer determinantal ideal of the matrix. This result has as a
corollary a theorem of Rushanan. We also study graphs having the same
determinantal ideals with at most one indeterminate; the socalled
codeterminantal graphs, which generalize the concepts of cospectral and
coinvariant graphs. We establish a necessary and sufficient condition for
graphs to be codeterminantal on $\mathbb{R}[x]$, and we present some
computational results on codeterminantal graphs up to 9 vertices. Finally, we
show that complete graphs and star graphs are determined by the SNF of its
distance Laplacian matrix.
