Student Research Profiles

Julian Sahasrabudhe (2011)	Niamh Chaparro (2010)	Aleks Vlasev (2010)	Bill Bao (2009)	Asif Zaman (2009)	Ryan Coghlan (2009)	Steve Melczer (2009)

Jacobi Determinant Identity Interpreted in the Form of Dodgson

Aleks Vlasev

Mathematics Major
Supervisor: Dr. Karen Yeats

Polynomials

Given a graph defined by N vertices, a spanning tree is a collection of N+1 edges of the graph (a subgraph) so that you can get from one vertex to any other vertex by travelling along edges of the tree. One can calculate the spanning tree polynomial by giving a variable name to each edge and taking into account all spanning trees of the graph. Karen Yeats considered a more general formula. In it one partitions some vertices into groups of the same "color". Then one has to consider all spanning forests (collection of distinct trees) such that each color is spanned by exactly one tree. One small consequence of this construction is a pretty formula relating a product between a tree polynomial and forest polynomial on three colors to a sum of products of polynomials of two colors. My research aimed to generalize this formula. The goal was to obtain a formula relating a product between a tree polynomial and a forest polynomial of four colors to a sum of products of polynomials of three colors.

I spent the first part of the summer finding such a formula. Once I found one, I spent the rest of the summer trying to prove that it is correct. The formulas available to me are true up to a sign, that is, the answer is at most off by a negative. The difficulty is in keeping track of all the sign changes that may potentially happen in the computation.