# Mathematics Colloquia and Seminars

Given a graph $G$ on $n$ vertices and a graph matrix $M_G$ related with $G$ and a commutative ring with identity $R$. Then the $k$-th determinantal ideal, denoted by $\mathcal{I}_k(M_G)$, is the ideal in $R[x_1,\ldots , x_n]$ generated by the k minors of the matrix $M_{G,X}$ whose $i$-th diagonal entry is $x_i$ and its off-diagonal entries coincide with those of $M_G$. In this talk, we will discuss some properties of determinantal ideals of graphs. Moreover, we will explore connections between this concept and other algebraic invariants of graphs, mainly with the sandpile group.