Codeterminantal graphs

We introduce the concept of codeterminantal graphs, which generalize the concepts of cospectral and coinvariant graphs. To do this, we investigate the relationship of the spectrum and the Smith normal form (SNF) with the determinantal ideals. We establish a necessary and sufficient condition for graphs to be codeterminantal on 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.