Algebraic Graph Theory-Ralihe Raul Villagran

Title: Determinantal ideals of graphs

Abstract: Let $A$ ($D$) denote the adjacency (distance) matrix of a graph $G$ with $n$ vertices. We define the $k$-th determinantal ideal of $M_X:=diag(x_1,x_2,\ldots ,x_n)+M$ as the ideal generated by all of its minors of size $k\leq n$. If $M=A$, we call this the $k$-th critical ideals of $G$. On the other hand, if $M=D$, we call it the $k$-th distance ideals of $G$. These algebraic objects are related to the spectrum of their corresponding graph matrices, their Smith normal form, and in consequence to their sandpile group for instance. In this talk, we will explore some of the properties and applications of these ideals.