Mathematical Sciences Department
Colloquium
Speaker: Ralihe Raul Villagran, WPI
Friday, October 6, 2023
11:00 am - 12:00 pm
HA209, Forkey Conference Room
Zoom Meeting ID: 938 8407 5985
Title: On some algebraic invariant of graphs: from their sandpile groups to determinantal ideals
Abstract: In this talk we will introduce some combinatorial and algebraic aspects of the sandpile group of a graph. In particular, we will see that the algebraic structure of the sandpile group of a graph G is characterized by the Smith normal form of its Laplacian matrix, L(G), more precisely by the diagonal elements of said Smith normal form, also known as the invariant factors. This fact leads us to define the determinantal ideals of the generalized Laplacian matrix and we will show a couple of examples on how these determinantal ideals are helpful to compute the algebraic structure of sandpile groups. Lastly, considering the spectrum and the invariant factors of several matrices defined by graphs, we ask the question: what algebraic invariants perform better when we want to distinguish graphs? and present some computational results.